PSI - Issue 80

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

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The stress-strain state is characterised by the small strain tensor ε ( u ) = 1 2 ∇ u + ( ∇ u ) and stress tensor σ which also defines the aforementioned surface tractions p = σ · n . The relation between them is influenced by elasto-plastic material parameters including the elastic sti ff ness C (generated for the isotropic material by the (plain strain) bulk modulus K p and the shear modulus µ ), yield stress σ y , and plastic hardening sti ff ness κ h . Permanent shape changes are characterised by a trace-free plastic strain tensor π so that the stress state in an undamageable material would be characterised by the stress tensor σ = C : ( ε − π ). The phase-field damage model resulting in a crack formation process is represented by an internal parameter α whith a damage-like character. An actual crack denoted Γ c in Fig 1 can be identified by a Gri ffi th-like energy ex pressed by an integral Γ c G c d Γ , with introduced parameter of fracture energy G c considered as a crack characteristic. Nevertheless, the integration domain Γ c is not known provided that the crack develops according to the loading his tory related to the (time) variable t . The internal parameter α available at each point of the material domain serves as a regularisation variable defining the state of the material relative to the crack state: α ∈ [0; 1] is defined so that α = 0 pertains to the pristine material and α = 1 reflects the actual crack. The values between introduce a smear crack region. The previous energy integral is replaced by a regularised one, as introduced in Ambrosio and Tortorelli (1990), to make displacements continuous across a crack, and having high gradients dependent on a length scale pa rameter . The regularisation based on Tanne´ et al. (2018) introduces the energy equivalent to that of the cracks in the form: Ω 3 8 G I c 1 α + ( ∇ α ) 2 d Ω . Additionally, the crack is represented by a degraded material: its strain energy (and as a result the elastic properties K p and µ ) progressively decrease according to a degradation function Φ obeying the relations Φ (0) = 1, Φ (1) = δ , (0 < δ 1 to guarantee positiveness of the energy in the case of a crack), Φ ( x ) > 0 for all x ∈ [0; 1] (for computational purposes also Φ (1) = 0 and Φ ( x ) > 0). Finally, the material is degraded within a band whose width is determined by , see Fig. 1. The assumptions about cracks and plastic deformations provide an expression for the stored energy as follows: E ( t ; u , π ,α ) = Ω Φ ( α ) K p sph + ( ε ( u )) 2 + µ | dev( ε ( u ) − π ) | 2 + K p sph − ( ε ( u )) 2 + 2 + (1) and where v ± = max(0 , ± v ). The elastic energy term is written in Eq. (1) using an additive orthogonal split of the strain tensor ε into its spherical sph ε and deviatoric dev ε parts. The split allows to define material degradation related to volumetric or shear strain independently, and to separate tensile ( + ) and compressive (-) parts of the spherical tensor in order not to degrade under compression. Simultaneously, it provides the relation for the current state of stress in the form σ = 2 Φ ( α ) K p sph + ( ε ( u )) + µ dev( ε ( u ) − π ) + 2 K p sph − ( ε ( u )). Summarising, the current state of the body at the instant t is described by three variables: the displacement field u and two internal parameters of the plastic strain π and of the phase-field damage α . Another ingredient to the energy balance is the energy of the external forces, given here by the boundary force functional F ( t ; u ) = Γ N f ( t ) · u d Γ . (3) Both processes which lead either to cracks or to permanent deformations dissipate energy. Thus, one part of dis sipation is caused by elasto-plastic character of the material. Reaching the yield surface, which is dependent on he predefined yield stress, dissipates energy proportional to changes in plasticised zones. Alternatively, under condition that the crack propagates in other than opening mode, there may appear additional energy dissipation. This phe nomenon can be simulated by introducing the mode dependent fracture energy, see e.g. Benzeggagh and Kenane (1996). Moreover, crack propagation is generally a unidirectional process. Having in mind the phase-field regulari sation, the internal parameter α should be constrained as ˙ α ≥ 0 in Ω . All these assumptions can be indicated by a 1 2 κ h | π | 3 8 G I c 1 α + ( ∇ α ) 2 d Ω , valid in an admissible state at the instant t expressed by the constraints u | Γ D = g ( t ), 0 ≤ α ≤ 1, (2)