PSI - Issue 80

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

505

5

(a)

g ( t )

x 2

x 1

25 25

150

100

150

(b)

Fig. 2. A beam exposed to an increasing displacement load (a), and characteristic form of the mesh.

k τ ) is rendered as argmin H 1 k

Therefore, the couple ( u k condition for calculating α k τ , π

τ ( u , π ). The inclusion in Eq. (7) 3 is in the same way seen as a

τ = argmin H 2 k

τ ( α ), with

τ

k τ ,α + τ · R u

α − α k − 1 τ

k τ ( α ) = E t

k ; u k

k − 1 τ

k − 1 τ ;0 ,

,

(9)

τ , π

, π

H 2

which uses the minimiser of the functional H 1 k

τ . The unidirectionality of the demage evolution requires the second

minimisation to consider constraints caused by the conditions in Eqs. (2) and (4) which read 0 ≤ α k τ ≤ α k − 1 τ .Within each time step, these two minimisations are solved to obtain the solution pertinent to the instant t k . The computational implementation approximates all state variables within the t range [0 , T ] by a finite element mesh and use them within an in-house MATLAB code of FEM analysis which uses implementation in the MATLAB environment described in ? . The code based on Eq. (1) provides with respect to deformation variables a quadratic functional so that the minimisation of (8) is based on quadratic programming (QP) algorithms. The minimisation of (9) generally leads to a non-necessarily quadratic though convex functional (based on what is assumed on the function Φ ( α )) for which the QP algorithm is applied sequentially. The functionality of the model is documented by analysing a constrained beam loaded by an increasing prescribed displacement g as shown in Fig. 2. The displacement is prescribed according to the time parameter t as g ( t ) = v · t for t ∈ [0 , 1 . 5] and v = 1mms − 1 and applied incrementally so that ∆ g = v · τ with various τ to see how the solution changes for refined discretisation. Simultaneously, the spatial mesh is refined proportionally, i.e. if the smallest mesh element size is denoted h , the ratio h /τ is kept constant. The mesh is irregular with refinements at the zone close to the load as shown in Fig. 2 for the roughest mesh used in the calculations. The characterisation of the material include elastic properties (introduced in Eq. (1) for an undamaged material): K p = 22 . 22GPa, µ = 13 . 33 GPa. Next, the elastic range is determined by the fixed yield stress σ y = 1MPa, the evolution of plastic zone is a ff ected be the strain hardening parameter κ h = K p / 100 and visco-plastic characteristic d = 0 . 1MPas − 1 . Finally in the model, the crack nucleation and propagation are controlled by the fracture energy G I c = 0 . 5Jm − 2 , G II c = 10 G I c and the phase-field length parameter set to = 2 mm. The PFM degradation function Φ is chosen in a power form: Φ ( α ) = (1 − α ) 3 + 10 − 6 , which provides critical stress (trace) for damage initiation at tr σ = 3 K p G I c ( − Φ (0)) = 2 . 36MPa. The presence of plastic deformations and also crack propagation can be identified in Fig. 3. The curves express total applied load at the zone of the prescribed displacements. Various discretisations are used in the graphic where the finest seems to be su ffi ciently accurate. The graphs reveal various phases of he solution. After the initial elastic part, plastic deformations appear identified by the changed slope of the function for a material with plastic hardening. Anyhow, further increase of load provokes cracks so that material is degraded and a relatively steep decrease of the force manifests the crack propagation. Due to compression, the cracks stop propagating, the final loading part corresponds to such a situation. All these phases can be also identified in the graphics below. The next graphics show the evolution of variables related to the both dissipative phenomena. Fig. 4 shows the dis tribution of internal variables (plastic strain in the part (a) and phase-field damage in the part (b)) at various instants of the loading process. Pertinent stress variables are then demonstrated in Fig. 5. The pictures in Fig. 4 also demonstrate 4. An example