Issue 75

P. Grubits et alii, Fracture and Structural Integrity, 75 (2026) 124-156; DOI: 10.3221/IGF-ESIS.75.10

where f denotes the yield function and   is the plastic multiplier, as schematically illustrated for the von Mises yield surface under plane stress in Fig. 1.

Figure 1: Schematic illustration of the associated flow rule for the von Mises yield surface under plane stress conditions.

Considering again the body introduced earlier, the total stress field   ij t  at time t can be expressed, in accordance with the constitutive relations presented in Eqns. (1) and (2), as the sum of a fictitious elastic stress component   el ij t  , corresponding to the elastic part of the total strain, and a time-independent residual stress component R ij  , which accounts for the irreversible effects of plastic deformation:     el R ij ij ij t t      (4)

  el ij t  , associated with the fictitious elastic stress

  el ij t  , is related to the stress field

The fictitious elastic strain field

through the linear elastic constitutive relation introduced in Eqn. (2):

  t

 

el

el ijkl kl C t 

ij 



(5)

It should be emphasized that Eqns. (2) and (5) refer to different stress fields: Eqn. (2) involves the actual stress   ij t  , while Eqn. (5) is defined in terms of the fictitious elastic stress   el ij t  . These are connected through the decomposition in Eqn. (4), where the residual stress R ij  does not vanish. This stress decomposition facilitates the evaluation of the complementary strain energy associated with the residual stress field, offering a scalar quantity to characterize inelastic behavior within the elasto-plastic framework. To further clarify this concept, consider the total complementary plastic work accumulated along a loading path from 0 t  to t   . This work serves as a meaningful measure for quantifying the extent of plastic deformation and evaluating the inelastic performance of the body under consideration. If a time-independent, self-equilibrated stress field R ij  can be identified such that the following yield condition is satisfied throughout the volume V at any time t   :     0 el R ij ij f t     (6)

  p W  is bounded from above by the expression:

then the total complementary plastic work

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