PSI - Issue 80

Roman Vodička et al. / Procedia Structural Integrity 80 (2026) 501 – 508 R. Vodicˇka / Structural Integrity Procedia 00 (2025) 000–000

504

4

dissipation pseudo-potential

R ( u , π ; ˙ π , ˙ α ) = Ω

c ˙ α

3 8

1 2

σ y sph( ε ( u )) | ˙ π | +

d | ˙ π | 2

G c ( u , π ) − G I

η d Ω

(4)

+

provided that the constraining inequality for ˙ α is satisfied. The fracture mode dependence expressed by the function as G c ( u , π ) guarantees that for an opening crack there will be no additional dissipated energy. The dependence of yield stress on spherical strain takes into consideration e ff ect of some plasticity theories, like Drucker-Prager, that yield surface depends on spherical stress (first invariant of the stress tensor). Pertinent functions can be chosen as follows:

+ ( ε ( u )) 2 + ε ( u ) | 2

K p sph

+ µ | dev( ε ( u ) − π ) | 2

G c ( u , π ) =

,

σ y ( s ) = σ y0 − ω K p | s | .

(5)

K p | sph

µ | dev( ε ( u ) − π ) | 2 G II c

+

G I c

The relations which control the quasi-static evolution of the system include force equilbrium and flow rules for the internal variables. Here, they may be represented by nonlinear inclusions with initial conditions ∂ u E ( t ; u , π ,α ) F ( t ; u ) , u (0 , · ) = u 0 ,

∂ π E ( t ; u , π ,α ) + ∂ ˙ π R ( u , π ; ˙ π , ˙ α ) 0, π (0 , · ) = π 0 = 0 , ∂ α E ( t ; u , π ,α ) + ∂ ˙ α R ( u , π ; ˙ π , ˙ α ) 0, α (0 , · ) = α 0 = 0,

(6)

with ∂ denoting the partial subdi ff erential (e. g. R is non-smooth at zero damage or plastic strain rates). The prime is used for (Gateaux) di ff erential. It is also assumed that the energy functional is separately convex with respect to strain variables and to damage variable. The initial values for the internal variables pertain to a state with no plastic deformation and no damage.

3. Numerical solution and computer implementation

For the computer implementation of Eq. (6), a time stepping procedure and spatial discretisation have to be intro duced. Some features of the implementations are described here. A staggered form is considered. It means that all functionals are decoupled with respect to the deformation vari ables, which include u and π , and the phase-field variable α . In the numerical scheme the decoupling provides ben eficial convexity of the functionals under restrictions, e. g. for the function E in Eq. (1) the restriction E ( t ; · , · ,α ) is convex and also E ( t ; u , π , · ) is convex at each instant t while keeping the retained variables constant. Within the load stepping, a fixed time step size τ is used to describe varying load obtained at the instants t k = k τ for k = 1, . . . , T τ and the state of the system in the instant t k is expressed by the triple u k τ , π k τ ,α k τ . Therefore, the relations from Eq. (6) have to be written for separate instants, using the approximation of the rates of the variables, generically denoted w , by the backward finite di ff erence, e.g. ˙ w ≈ w k τ − w k − 1 τ τ . Next, the di ff erentiation with respect to the rates is substituted by di ff erentiation with respect to w k τ at the instant t k toobtain

, 0 f k

R u

π k

k − 1 τ

E t

k − 1 τ + ∂ v k τ k − 1 τ + τ · ∂ π k τ k τ + τ · ∂ α k τ

τ − π

k ; u k

k τ ,α

k − 1 τ

k − 1 τ ;

0 τ = u 0 ,

u

τ ,

τ , π

, π

∂ u k τ

τ

R u k − 1 τ

, 0 0, π 0

π k

k − 1 τ

E t k ; u k

τ − π

k τ ,α

k − 1 τ ;

(7)

τ = π 20 ,

τ , π

, π

∂ π k τ

τ

R u k − 1 τ

k − 1 τ τ

α k

E t k ; u k

τ − α

k τ ,α

k − 1 τ ;0 ,

α 0

τ = α 0 .

0,

τ , π

, π

∂ α k τ

Keeping a part of variables constant in the staggered approach renders a variational structure to the solved system. The inclusions in Eqs. (7) 1 and (7) 2 can be seen as minimisation conditions for the following convex functional

, 0 − F ( t k ; u ).

τ + τ · R u

k τ ( u , π ) = E t

π − π k − 1 τ τ

k ; u , π ,α k − 1

k − 1 τ

k − 1 τ ;

(8)

, π

H 1