PSI - Issue 52

Roman Vodička et al. / Procedia Structural Integrity 52 (2024) 242 – 251

247

6

R. Vodicˇka / Structural Integrity Procedia 00 (2023) 000–000

(a)

(b)

0 . 4

g ( t )

1

g ( t )

Fig. 2. Simple configuration (a) exposed to a displacement pulse (b).

τ + τ · R u

, 0

k τ ( u , e 2 ) = 2 · E t k ;

k − 1 τ

k − 1 τ

u + u k − 1 τ 2

u − u k − 1 τ τ

e 2 + e 2

e 2 − e 2

,α k − 1

k − 1 τ

k − 1 τ ;

,

,

,α

H 1

2

τ

(11)

+ 2 τ 2 · K

−

u k

k − 1 τ − τ v

k − 1 τ

τ − u

F ( t k ; u ).

τ 2

k τ ) is rendered as argmin H 1 k

Therefore, the couple ( u k

τ , e 2

τ ( u , e 2 ). In a similar way, the last inclusion in Eq. (10) is

τ = argmin H 2 k

observed as a condition for finding α k

τ ( α ), where

k τ ( α ) = 2 · E t

2

+ τ · R u

k − 1 τ τ

α k

α + α k − 1 τ

τ − α

k − 1 τ

k − 1 τ ;0 , 0 ,

k ; u k

k τ ,

τ , e 2

,

(12)

,α

H 2

which uses the result of the previous minimisation of the functional H 1 k τ . Thus, within each time step these two min imisations are resolved to get the solution pertinent to the time instant t k . It should be noted that the unidirectionality of the degradation process equips the second minimisation with the constraint provoked by the conditions in Eqs. (2) and (5) which can be written in the form 0 ≤ α k τ ≤ α k − 1 τ . In the computational implementation, all variables which define the trajectory within the time range [0 , T ] are approximated by an adequate finite element mesh and interpreted within an inhouse MATLAB code of FEM analysis which uses simple implementation in the MATLAB environment originally stemmed from Alberty et al. (2002). Based on Eqs. (1) and (3), it is generally admitted that with respect to deformation variables a quadratic functional is obtained for whose minimisation quadratic programming (QP) algorithms are applied, based on Dosta´l (2009). The other restriction generally leads to a convex functional (based on the aforementioned assumptions on the function Φ ( α )) for which the QP algorithm is applied sequentially. Crack nucleation is observed under a time dependent loading for a simplified domain shown in Fig. 2. The boundary conditions make the problem to depend only on x 2 and making a pulse displacement load as also shown in the same graphics, provides a wave spreading in the vertical direction and causing rupture at the center of the domain where both waves meet. The initial elastic properties (introduced in Eq. (1) for an undamaged material) are: K p = 2MPa, µ = 1MPa, the mass density is ρ = 0 . 75 gmm − 3 . These parameters cause the P-wave to propagate at the velocity of 2 m s − 1 . The hard-device loading g ( t ) according to the graph in Fig. 2 is applied with a refined time step of 2 µ s. The fracture energy in the domain is G c = 10Jm − 2 and the phase-field length parameter is set to ε = 10 µ m. The PFM degradation function Φ is chosen in a simple quadratic form: Φ ( α ) = α 2 + 10 − 6 . The mesh is regular, made of square elements of the size h = 10 µ m. The results study the propagation of wave inside material at the velocity given by the material parameters and its influence on degradation of the material as can be seen in Fig. 3 where strain energy density of the P-wave ω P is shown. The fourth time instant ( t = 0 . 276 ms) pertains to the moment, when the waves bumped each other. For a demonstration purpose, a modification of material characteristics is made in relation to described rheology: the inner sti ff ness ratio is γ = 1, the relaxation times are basically set to zero, changes are specified in the captions of graphics. Non vanishing viscous parameters cause slight attenuation of the waves. Nevertheless, when the waves meet at the center of the domain they trigger degradation and result in a crack placed here. Though, in the most damped case (the 4. Examples