PSI - Issue 52

Roman Vodička et al. / Procedia Structural Integrity 52 (2024) 242 – 251 R. Vodicˇka / Structural Integrity Procedia 00 (2023) 000–000

246

5

where ∂ denotes the partial subdi ff erential (e. g. R may jump for zero damage rates). It is also supposed that the energy functional (with respect to separated variables of the trajectory) is separately convex, the prime denotes (Gateaux) di ff erential. The initial value for the phase-field parameter pertains to a non-degraded state. The system (8) includes due to the inertial term a second order derivative with respect to t . In computations, it is sometimes useful to reduce it to a first order system by introducing the velocity as an independent variable by putting v = ˙ u . The system then changes to the following

˙ u = v ,

u (0 , · ) = u 0 ,

d d t K ( v ) + ∂ u E ( t ; u , e 2 ,α ) + ∂ v R ( u ,α ; v , ˙ e 2 , ˙ α ) F ( t ; u ) , v (0 , · ) = v 0 , ∂ e 2 E ( t ; u , e 2 ,α ) + ∂ ˙ e 2 R ( u ,α ; v , ˙ e 2 , ˙ α ) 0, e 2 (0 , · ) = e 20 , ∂ α E ( t ; u , e 2 ,α ) + ∂ ˙ α R ( u ,α ; v , ˙ e 2 , ˙ α ) 0, α (0 , · ) = α 0 ,

(9)

which will be also used in the computational model below.

3. Numerical solution and computer implementation

The evolution described in Eq. (9) requires to introduce a time stepping procedure to resolve the problem in time and within each time step to implement spatial discretisation. Some features of the implementations are described within this section. First, realise that all functionals can be decoupled respectively to deformation variables, which include u and e 2 , and the phase-field variable α . Thus also the numerical scheme may benefit from such decoupling, as the functionals then remain convex with respect to the two separated groups of variables, e. g. for the function E in Eq. (1) holds that E ( t ; · , · ,α ) is convex and also E ( t ; u , e 2 , · ) is convex at each instant t while keeping the remained variables constant. The computational scheme for the time discretisation then can be advantageously written in a staggered form. Additionally, as documented in Roub´ıcˇek and Panagiotopoulos (2017), the time stepping scheme for the first order system (9) may use the Crank-Nicolson formula, say, for eliminating numerical attenuation and having only the physical one caused by the used rheological model. Simultaneously, this formula also uses a mid-point calculation as it is used in the staggered approach. Therefore, a fixed time step size τ is used to describe time stepping obtained at the instants t k = k τ for k = 1, . . . , T τ and the state of the sytem in the instant t k is expressed by the triple u k τ , e k 2 τ ,α k τ . Themid point values are naturally distinguished by the half index and, generically for w , they are defined as w k − 1 2 = w k τ + w k − 1 τ . Therefore, the relations from Eq. (9) have to be written for separate time instants, using the approximation of the rates of the variables by the backward finite di ff erence, e.g. ˙ w ≈ w k τ − w k − 1 τ τ . The di ff erentiation with respect to the rates, generically ˙ w , is substituted by di ff erentiation with respect to w k τ at the instant t k .We obtain u k τ − u k − 1 τ τ = v k − 1 2 τ , u 0 τ = u 0 , K v k τ − v k − 1 τ τ + 2 · ∂ u k τ E t k ; u k − 1 2 τ , e 2 k − 1 2 τ ,α k − 1 τ + 2 · ∂ v k τ R u k − 1 τ ,α k − 1 τ ; v k − 1 2 τ , e 2 k τ − e 2 k − 1 τ τ , 0 f k τ , v 0 τ = v 0 , (10) τ 2

R u k − 1 τ R u k − 1 τ

, 0 0,

k − 1 τ + τ · ∂ e 2 k − 1 2 τ + τ · ∂ α k τ

E t k ; u

k τ − e 2

k − 1 τ

e 2

k − 1 2 τ

k − 1 2 τ

k − 1 2 τ

0 τ = e 20 ,

k − 1 τ ; v

, e 2

e 2

2 · ∂ e

,α

,α

,

k τ

k τ

2

τ

k − 1 τ τ

E t k ; u k

α k

τ − α

k τ ,α

k − 1 τ ;0 , 0 ,

α 0

τ , e 2

2 · ∂ α k τ

τ = α 0 .

0,

,α

The structure of the relations in Eq. (10) enables to formulate the problem in a variational manner. It requires some slight modifications of the system. First, by means of Eq. (10) 1 the variable v k τ can be eliminated from the system by substituting v k τ = 2 τ u k τ − u k − 1 τ − v k − 1 τ and replacing the di ff erentiation with respect to v k τ by that with respect to u k τ . Then the inclusions in Eqs. (10) 2 and (10) 3 can be seen as minimisation conditions for the following convex functional