PSI - Issue 48

Udaya B Sathuvalli et al. / Procedia Structural Integrity 48 (2023) 207–214

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Sathuvalli and Suryanarayana/ Structural Integrity Procedia 00 (2019) 000–000

Fig. 3. Evolution of triaxial stress distribution in elastic perfectly plastic cylinder cross section with progressive plasticization

The structural limit of an elastic-perfectly-plastic cylinder is determined by calculating the load parameters that plasticize the entire cross section, i.e. when the radius of the elastic-plastic boundary ρ equals the outer radius b . The ordinate in Fig. 3a shows the triaxial stress (Eq. (21) in Appendix A) at each stage of plasticization. In this example, the cylinder is loaded by increasing the axial strain and holding the pressures constant. The orange curve in Fig. 3b shows the axial strain as a function of the radial location of the elastic plastic boundary. The black curve in Fig 3b shows the normalized axial force. The axial force that can be supported by the cylinder cross section reaches a maximum (denoted by F z,fp ) when ρ / a = 1.24, and the axial strain (denoted by ε z,fp ) is 2.215 x10 -3 . This represents the structural limit of this tube for the indicated set of parameters. As expected, the normalized triaxial stress in the plasticized zone is unity in an elastic perfectly plastic material. Fig. 4 shows the loci of the structural limits for three thick-walled casings with zero internal pressure. The ordinate in both panels of the figure represents the external pressure normalized with respect to the pressure p c,y required to yield the cylinder ID when the axial load (or strain) and internal pressure are both zero. This external pressure is given by ( ) , 2 2 1 c y yp p γ σ γ − = (3) where o d t γ = . (4) In Fig. 4a, the abscissa (of the three colored curves) represents the axial strain when the cross section cannot support further loading (the orange circle in Fig. 3b). The abscissa in Fig. 4b is the corresponding normalized axial load (the black dot in Fig. 3a). Each point of a given locus in Fig. 4 is obtained from a numerical solution of the equations in Appendix A. These equations describe the response of the cylinder (Fig. 2) with a constrained plastic zone. It is important to recognize that the loci in Fig. 4 assume that axial strain is monotonously increased after a given differential pressure is applied. For example, if the normalized external pressure is 0.8 and the cylinder d o / t = 8 (green curve in Fig. 3a), the axial strain that causes structural instability is 0.734 ε yp . The corresponding axial load is 0.453 F yp (green curve in Fig. 3b). In terms of practical application, the limiting axial strain is compared with the geomechanical strain transferred by the formation to the casing. The compression and tension, respectively, in the reservoir and the overburden are poroelastic consequences of pressure depletion in the reservoir. In reality, it is difficult to determine if the pressurized casing is subjected to axial geomechanical strain, or a strained casing is pressurized. Fortunately, despite plasticization, the ultimate structural limit of the casing is relatively insensitive to sequence of loading (i.e., pressure followed by tension or vice versa, Madhavan et al. 1993).