PSI - Issue 48

Udaya B Sathuvalli et al. / Procedia Structural Integrity 48 (2023) 207–214 Sathuvalli and Suryanarayana/ Structural Integrity Procedia 00 (2019) 000–000

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in Fig. 1. During field wide design and root cause analyses of failures, parametric sensitivity calculations are needed. The methods and results presented here can be obtained from a numerical solution of the governing equations described in the appendices. The numerical solutions can be implemented in macro-enabled spreadsheets with little computational overburden. 5. Acknowledgements The authors thank the management of Blade Energy Partners for providing the resources to prepare this paper. The first author is grateful to the late Dr Paul Paslay of Manatee, Inc. for providing guidance in formulating this problem. 6. Appendix A: Governing Equations for the Thick-Walled Cylinder The solution presented here closely follows the method presented by Prager and Hodge (1951) for the plasticization of an internally pressurized cylinder in plane strain. The solution begins by using Eq. (1) to determine the parameters ( P int , P ext , ε z ) that yield the cylinder ID. An increment in any of these three variables increases the area of the axisymmetric plastic zone, and moves the location of the elastic-plastic boundary toward the cylinder OD. The solution proceeds by calculating the location of the elastic-plastic boundary for a load increment. Qualitatively speaking, there are three main aspects to the solution- 1) the stresses and strains in the elastic zone, ( ρ ≤ r ≤ b ) determined by the Lame equations and Hooke’s law; 2) the stresses and strain rates in the plastic zone, ( a ≤ r ≤ ρ ) related by the Prandtl-Reuss relations, 3) the radial stresses at the cylinder ID and OD determined from the applied pressures, and the continuity of radial displacement and stress across the elastic-plastic boundary. The equations for mechanical (radial) equilibrium and the material hardening rule provide closure to the equations. The stresses in the elastic zone (Fig. 2; ρ ≤ r ≤ b )are given by the Lame equations (Boresi et al. 1993). The stresses and strains in the plastic zone (Fig. 2; a ≤ r ≤ ρ ) are determined by the Prandtl-Reuss relations (Prager and Hodge, 1951, pp. 27-31) ( ) 1 3 2 r r z r z du E dr θ θ ε σ νσ νσ σ σ σ λ = = − − + − −       (12) ( ) 1 3 2 z r z r u E E r θ θ θ ε σ νσ νσ σ σ σ λ = = − − + − −       (13) ( ) Independent of = 2 E r ε σ νσ νσ σ σ σ λ = − − + − −      (14) In the above equations, the dot above a symbol denotes the temporal derivative, and λ is the plasticity parameter greater than zero. For an elastic-perfectly plastic material, this parameter is unknown and must be calculated as part of the solution. For an isotropically hardening material, this parameter is given by ( ) ( ) ( ) 2 3 2 2 2 4 ' r z r z r z r z VME E H θ θ θ θ λ σσσσ σσσσ σσσσ σ = − − + − − + − −         (15) where H’ is the slope of the true stress versus plastic strain curve, and ( ) ( ) ( ) 2 2 2 σ σ σ σ σ σ σ   = − + − + − (16) For an elastic-perfectly plastic material, the von Mises equivalent stress is constant in the plastic region, and equal to the uniaxial yield stress in tension. This implies that, ( ) ( ) ( ) 2 2 2 0 r z r z r z r z θ θ θ θ σσσσ σσσσ σσσσ − − + − − + − − =    (17) The Prandtl-Reuss equations relate the rate of change of plastic strain deviation to the instantaneous stress deviation. Finally, the radial equilibrium equation for the cylinder is given by 0 r r d dr r θ σ σ σ − + =    (18) Multiplying Eq. (18) by r, and the differentiating both sides with respect r , and equating the right hand sides of the resulting equation to right hand side of Eq. (17), we obtain ( )( ) ( ) ( ) 1 2 2 0 3 3 z r r r z r z r d d d r r r dr dr dr d d d r d r dr dr dr dr θ θ θ θ θ σ σ σ σ σ ν λ σ σ ν ν σ σ σ λ λ σ σ σ − + + − + − − +   − − + − − =             (19). Eqs. (13), (14), (19), (18) and (15) or (17) provide five equations for the five unknown variables σ r , σ θ , σ z , λ , u in the plastic zone and their temporal and spatial derivatives. 1 3 z z r z r θ θ 1 2 VME r z z r θ θ   .