PSI - Issue 68

94 4 Nhan T. Nguyen et al. / Procedia Structural Integrity 68 (2025) 91–98 N.T. Nguyen et al. / Structural Integrity Procedia 00 (2025) 000–000 ,- $ > / ; =$ $ and = ,- $ "$ , = " 1 $ are functions of stresses and internal variables of which form is chosen such that the dissipation rates are independent of stress. Here, the non-dimensional and non negative parameters + , # $ and & $ are introduced that represent the three aspects of dissipative mechanisms related to damage, volumetric and shear deformation, respectively. Note that the condition + + # $ + & $ =1 is applied in this current development state to demonstrate the unity of the whole dissipation budget produced by the model. From the above set of generalised dissipative stresses and the dissipative rates, according to the thermodynamics framework for hyperplasticity, the Legendre transformation is utilised to obtain the yield function in mixed stress energy space, ∗ . Details can be found in Nguyen et al. (2024). The yield function in true stress space, , is obtained consequently: = ,- !$ @(+) + "(!/+) % (%2),- #$ A(!/+)>"@(+)(%2)BC〈$〉F % −1≤0 (9) where = (1 − ) G + H is the frictional slope in triaxial stress space. The composition of the damage energy term ̅ + = ̅ #+ + ̅ &+ represents the energies induced by volumetric and shear stress, that conjugate to the total damage. The damage evolution function ( ) = I ' % "J K!B %*+,(-)$'.%()'% L % M(!/+)B %*+,(-)$'.%()'% N % includes ℎ being the crackband thickness, O being the uniaxial tension strength, @ being the fracture energy over the fractured surface. The flow rules, in their incremental form, are: # $ = (P ∗ (, ! & = 2 ; ! & $ − ̇ − ̇ & $ (10) & $ = (P ∗ (, # & = 2 ; # & 1/<$ (11) = (P ∗ (, $ = 2 ; $ ,- $ /=$ (12) In the above equations, ∗ is the yield function in generalised stress space, equivalent to a plastic potential in classical plasticity theory, and + , # $ , and & $ are dissipative generalized stresses. From the definition of the dissipation potential, Eq. (5), and the form of flow rules, Eqs. 10-12, we can formulate the total dissipation budget ( Q ) and the dissipation ratios from each mechanism ( # $ , & $ , + ) as: Q = Q + + Q # $ + Q & $ = 4 + + 4 & $ + 2 * # $ − & $ − + , = 2 (13) # $ = 45 ! & 45 = # $ − & $ − + (14) & $ = 45 # & 45 = 2 & $ (15) + = 45 $ 45 = 2 + (16) in which # $ = : $ ; ! & , & $ = 1 : / ; <$ # & , + =