PSI - Issue 82

M.S. KannanKulavan et al. / Procedia Structural Integrity 82 (2026) 44–50 KannanKulavan et al. / Structural Integrity Procedia 00 (2026) 000–000

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2.2. Weight Function Formulations

The weight function (WF) is a geometry-specific construct that acts as a Green’s function for cracked bodies. It enables the evaluation of fracture parameters by integrating the uncracked stress field with a pre-characterized response function. Originally developed for SIF calculations, the WF framework is equally applicable to T-stress evaluation Fett (1997). This approach avoids repeated full-field simulations by decoupling geometry from loading. The weight function (WF) method enables the evaluation of fracture parameters by integrating the stress field of the uncracked body with a geometry-specific kernel. For Mode I loading, the stress intensity factor K I is given by: K I = a 0 σ ( x ) m ( x , a ) dx (1) where σ ( x ) is the normal stress acting on the crack plane in the uncracked configuration, a is the crack length, and m ( x , a ) is the Mode I weight function. This formulation was introduced by Bueckner and later formalized by Rice Rice (1972). The WF for the K I of a double edge cracked circular ring was derived by the authors using two reference loading configurations K.K. and Surendra (2025a). Similarly, the T-stress can be expressed using a second-order weight function m T ( x , a ) as: T = a 0 σ yy ( x ) m T ( x , a ) dx − ( σ yy − σ xx ) | x = a (2) This approach was extended by Fett Fett (1997), who demonstrated that m T ( x , a ) can be constructed using two reference crack-face loading configurations: uniform and linearly varying pressure. These form a complete basis for symmetric tractions and allow the derivation of a semi-analytical WF for T-stress evaluation. Weight function for T-stress is given by Eq. 3 K.K. and Surendra (2025b). m T ( x , a ) = C 1 1 − x a 2 + C 2 1 − x a 1 / 2 (3) where σ xx and σ yy are the Cartesian stress components in the uncracked body, a is the crack-length, m T ( x , a ) is the weight function for T -stress.

where, C 1 = − 4

a (3 V 1 − 5 V 2 ) and C 2 = 5

2 a (3 V 1 − 4 V 2 ); V 1 and V 2 are the normalised T -stress from reference configu

rations obtained from K.K. and Surendra (2025b).

2.3. Lame´ Solution for the Uncracked Annulus

To support the weight function integration, the classical Lame´ solution for thick-walled cylinders subjected to internal and external pressure was reformulated for parametric analysis. For a cylinder with inner and outer radii R i and R o , and corresponding pressures P i and P o , the normalized stress expressions were derived in terms of the pressure ratio η = P o / P i and the normalized radius ¯ r = ( r − R i ) / ( R o − R i ) as in Eqs. (4) Srinath (2016). These relations enable systematic variation of both η and the geometry ratio R i / R o , providing a scalable framework for evaluating stress gradients in the uncracked annulus, which serve as input for weight function–based T-stress calculations.