PSI - Issue 68

Nhan T. Nguyen et al. / Procedia Structural Integrity 68 (2025) 91–98 N.T. Nguyen et al. / Structural Integrity Procedia 00 (2025) 000–000

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2. Constitutive model formulation 2.1. A coupled damage-plasticity model controlling energy dissipation

The key feature of the proposed thermodynamics-based continuum framework based on the work of Houlsby and Puzrin (2000, 2006) is the capability to have a complete description of a constitutive model using two explicitly defined energy potentials, following a set of well-established procedures. Hence, the details definition and meaning of the upcoming thermodynamics-related term can be found in Houlsby and Puzrin (2000, 2006). In this paper, only the essential equations needed for highlighting the dissipation control feature and the fundamental components of a constitutive model will be provided, the details of thermodynamics formulation can be found in Nguyen et al. (2024). The Helmholtz free energy potential is: = ! " (1− ) * # − # $ , " + % " (1 − ) * & − & $ , " + ! " 〈 # − # $ 〉 " (1) where and are the elastic moduli of the material related to the bulk and shear terms, respectively. The total volumetric strain, # , and shear strains, & , are decomposed into the plastic part and the elastic part, noted with the superscript being either " " or " " ; thus, the elastic volumetric strain can be written as #' = # − # $ , and the elastic shear strain as & ' = & − & $ . The Macaulay operation 〈 〉 = 5 0 , , <≥ 0 0 helps activate/deactivate the extra “bulk” term in the equation that represents the material having its bulk stiffness degrading under tension while fully recovering from damage, , when is under compression. From this specific form of free energy, the following forms of true stresses can be obtained: = ( ( * ) !" = (1 − ) #' + 〈 #' 〉 = : #' (1 − ) #' ( , #' >0) ( , #' <0) (2) = ( ( * ) #" = 3(1 − ) & ' (3) ̅ + =− ( ( ) + = ! " #'" − ! " 〈 #' 〉 " HIIIIJIIIIK ,- !$ + % " & '" HJK ,- #$ =L "(!/ ! +) % 1 % %2 "(!/ ! +) % M $ % 3 + 1 % %2 N ( , #' >0) ( , #' <0) (4) where is the mean stress, is the deviatoric stress, and ̅ + is the generalised stress related to damage energy. The dissipation potential, which is homogenous first-order in terms of plastic volumetric strain, ̇ # $ , plastic shear strain , ̇ & $ and damage, ̇ , assumes the following form that automatically satisfies the second law of thermodynamics: Q = ̅ # $ ̇ # $ R 45 ! & + ̅ & $ ̇ & $ R 45 # & + ̅ + R ̇ 45 $ = S( + ) " + * # $ , " + * & $ , " ≥0 (5) where ̅ # $ , ̅ & $ and ̅ + are generalised dissipative stresses that form into dissipation terms, ΦQ # $ , ΦQ & $ and ΦQ + , as a product with plastic variables. The functions φ 6 , φ 7 8 and φ 9 8 governing the plastic volumetric, plastic shear and damage dissipation rates are: # $ = # $ * ̇ # $ + ̇ + ̇ & $ , (6) & $ = & $ ̇ & $ (7) + = + ̇ (8)