Longitudinal boundary strain

For the boundary value d = 2 (brittle deformation) let us obtain from the Eq. (14.7) the greatest value = 0.333. In its turn, the longitudinal boundary strain 8 can be estimated as follows [15] ... 

The exact calculation of radial and circumferential stresses in each layer requires the solution of N+2 linear equations in N-r2 unknowns, namely the N-H radial displacements of the layer boundaries, and the longitudinal strain of the vessel. We simplify by assuming that the internal pressure is applied only to the steel shell, and that the other layers follow the expansion of the steel. We also assume a condition of plane stress that is, no stress in the axial direction of the cylindrical vessel. We also consider the layer as being flat when layer stresses are being computed. 

As mentioned previously, the boundary condition in DD and FE are different. Periodic boundary condition is used in DD analysis to take into account the periodicity of single crystals whereas confined boundary condition is used in the FE analysis to achieve the uniaxial state of strain. In order for the boundary conditions in FE and DD to be consistent, periodic FE boundary condition is implemented as well. This implementation of periodic FE boundary condition yields a relaxed state of stress with low peak pressure when compared to the experiment as illustrated in Fig. 9(a). Furthermore, both shear and longitudinal waves are generated which is discordant with plane wave characteristics as shown in Fig 9(b). Fig 10 shows the deformed shape when confined and periodic boundary conditions are used. In the confined case there is no distortion in the RVE. However, for the periodic case, considerable... 

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