PSI - Issue 80

Alok Negi et al. / Procedia Structural Integrity 80 (2026) 203–211 AlokNegi / Structural Integrity Procedia 00 (2023) 000–000 3 where C e is the fourth-order elastic sti ff ness tensor, and total strain ϵ is additively decomposed into elastic and plastic parts. The degradation function g e ( ϕ ) and g p ( ϕ ) modulates the elastic energy and plastic energy, 205

g e ( ϕ ) = (1 − ϕ ) 2

+ k and g p ( ϕ ) = β g e ( ϕ ) + (1 − β )

(3)

where β = 0 . 1 is a weighting factor for the plastic contribution [Cupertino-Malheiros et al. (2024)] and k ≪ 1 ensuring numerical stability. Plastic behavior is captured using classical J 2 plasticity with isotropic hardening, where stored plastic energy density is given by,

σ y 0 ε 0 ( n p + 1)     1 +

− 1  

ε p ε 0 

= σ y 0  1 +

ε p ε 0 

n p + 1

n p

p )

 , with σ f ( ε

(4)

ψ p =

where the flow stress evolves with equivalent plastic strain ε p with σ

y 0 as the initial yield stress, n p the strain-hardening

exponent, and ε 0 = σ

y 0 / E the initial yield strain. Minimizing the potential energy Π leads to the governing equations.

Mechanical equilibrium is expressed as,

p = 0 , where σ = g e ( ϕ ) C

e : ( ϵ − ϵ

∇· σ + b

p )

(5)

where σ is the Cauchy stress tensor and b p are body forces. The phase-field evolution equation reads,

ϕ l − l ∇ ϕ ·∇ ϕ  − 2(1 − ϕ ) H = 0 , where H ( x , t ) = max τ ∈ [0 , t ]  ψ +

G c ( C ) 

e ( x ,τ ) + βψ p ( x ,τ ) 

(6)

where H is a history field that ensures irreversibility of damage with ψ + e denoting the tensile elastic strain energy density obtained from volumetric-deviatoric decomposition [Amor et al. (2009)]. Boundary conditions include pre scribed displacements or tractions for the mechanical problem, and a homogeneous Neumann condition ( ∇ ϕ · n = 0) for the phase field, ensuring no damage flux across the boundaries.

2.2. Hydrogen transport and fracture toughness degradation

The mass conservation equation governs the evolution of mobile hydrogen concentration C ,

D app C ¯ V H

∂ C ∂ t

d

= 0 , where J d

+ ∇· J

= − D app ∇ C +

(7)

RT ref ∇

σ H

where J d is the di ff usion flux with D app the apparent di ff usivity, ¯ V H the partial molar volume of hydrogen, R the universal gas constant, and T ref the reference temperature. Surface exposure to sour environments is modeled by prescribing a surface concentration C = C b ( t ), calibrated from experimental hydrogen permeation data. Hydrogen accumulation leads to degradation of fracture toughness, modeled via an exponential degradation law, G c ( C ) = G c 0  G min c G c 0 +  1 − G min c G c 0  exp( − α C )  (8)