Issue 47

A. Chouiter et alii, Frattura ed Integrità Strutturale, 47 (2019) 30-38; DOI: 10.3221/IGF-ESIS.47.03 31 I NTRODUCTION he expansion bellows, which permit to compensate the difference in expansion between the tubular bundle and the shell in a fixed tube sheet heat exchanger, is a structural component formed by one or more convolutions. The expansion joint is an integral part of a heat exchanger. It must simultaneously ensure the flexibility due to thermal expansion and withstand the internal pressure in the heat exchanger. In addition, it must be sufficiently flexible longitudinally to accommodate the deviations for which it is designed and must absorb the regular or irregular extension generated by the gradients temperature and pressure. In the literature, we find a wide variety of references related to the calculation of expansion bellows cyclic life. We can cite as an example: Lee [2] has used a finite element analysis technique applied to the parameters of the bending and folding process of the metal bellows. Becht IV [3] has shown that the fatigue behavior of the bellows can be well predicted by dividing the bellows fatigue data on the basis of geometry parameters. Zupei [3] has used approximate calculation methods, which he compared to the results obtained by the finite element method for the U-shaped bellows. Broman et al. [5] determined the dynamic characteristics by manipulating the finite element parameters of the bellows beam. Kang et al. [6] have studied the process of forming different types of tubular bellows using a single stage hydroforming process. Faraji [7, 8] has experimentally and numerically analyzed the optimal parameters for the manufacture of metal expansion bellows. Bo-wun Huang et al [9], in their work, the dynamic properties of coupled tube-array structures with the axial loads are investigated in heat exchanger. Tingxin et al. [10] have experimentally studied the behavior of toroidal bellows compared to U-shaped bellows, they showed that toroidal bellows have a lower stress induced by internal pressure, longer fatigue life, greater ability to resist to instability of internal pressure and are more suited to situations of higher internal pressure. The Code ASME VIII div 1 [11] is widely the most comprehensive code that deal with the design of expansion bellows, their calculations as well as their standards. In this work, we develop a method based on the concept of continuum damage mechanics (CDM), which reaches a stage of maturity allowing modeling any type of degradation. The life of the expansion bellows is obtained by using a post processor based on the of continuum damage mechanics using Newton's iterative method which is unconditionally stable and ensures good convergence. In general, the damage is much localized and occupies a small volume, ie a representative volume element RVE compared to macroscopic scale of the structure. This is due to the high sensitivity to microscopic damage concentrations. Benzerga [12] noted that the damage effect on stress conditions occurs only in a very small area of the material. In other hand, the coupling damage behavior can occur only in RVE. It is the principle of the analysis of the local coupling damage-deformation presented by Lemaitre and Doghri [13]. Our approach can be divided into two stages: In a first step, the finite element calculation code (ANSYS) will allow in elasticity or elasto-plasticity to obtain the critical region of the structure (M*) where the Von Mises stress is maximal. In a second step, a post processor based on the Continuum Damage Mechanics concept using the Newton iterative method was applied at this critical point (M*) for the determination of the critical lifetime of the expansion bellows under service conditions of pressure and thermal expansion. This lifetime is the value corresponding to the critical damage value Dc (crack initiation). This method was validated by comparing our numerical results with those of ASME code [11]. Figure 1 : Locally coupled analysis of crack initiation. T