PSI - Issue 80

Roman Vodička et al. / Procedia Structural Integrity 80 (2026) 501 – 508

502

2

R. Vodicˇka / Structural Integrity Procedia 00 (2025) 000–000 f g Γ N

Γ D

Ω

s

n

x 2

Γ D

Γ c

2 ℓ

g = 0

x 1

0

Fig. 1. Description of a deformable body, a smear crack characteristic, boundary conditions and constraints.

of the smear crack can by identified with some characteristic material length and a ff ects the cracking process as dis cussed by Tanne´ et al. (2018); Sargado et al. (2018); Wu (2017). The properties of materials and form of crack patterns appearing under various loading conditions finally result in initiation and propagation of cracks in various modes. Distinguishing the fracture energy in shearing and opening modes was addressed a few newer approaches of PFM, e.g Zhang et al. (2017); Wang et al. (2020); Feng and Li (2022). Anyhow, plastic deformations contribute to dissipation of energy and naturally a ff ect cracking in various crack modes. The formulation of phase-field fracture model for ductile materials appeared in Duda et al. (2015); Ambati et al. (2015); Borden et al. (2016). Naturally, those models defined the competition between the processes of plasticity and rupture. Simultaneously, it expects to make di ff erence between tensile and shearing deformation in cracks arising in mixed fracture modes and making sense of combining the stress-strain relations also with plasticity, as documented in Huber and Asle Zaeem (2023). Such a combination of inelastic processes is intended for the model in the present contribution, too. The principal objective of the paper is to extend the previous author’s model for quasi-brittle materials introduced in Vodicˇka (2023, 2024) to the ductile materials where plastic strain may evolve prior to a crack initiation which naturally considers also dependence on the fracture mode. The following sections demonstrates mutual influences of both phenomena. First, the computational model is proposed, and second, illustrating by a simple material element it intends to provide reasonable data.

Nomenclature

u ε

[m] displacement vector [-] small strain tensor

[Pa] stress tensor

σ α

[-] phase-field damage parameter

[-] plastic strain tensor

π

K p

[Pa] plain strain bulk elastic modulus

[Pa] shear elastic modulus

µ

[Pa] yield stress

σ y

[Pa] plastic hardening sti ff ness

κ h

2 ] fracture energy

G c [Jm −

[m] phase-field length-scale parameter

2. Description of the computational model

Consider a structural element represented by a domain Ω made of a elasto-plastic material which includes also kinematic hardening and a simple rheology related to plasticity introducing a little rate-dependence into the model. The outer contour Γ of the domain is split into two non-overlapping parts Γ D and Γ N according to prescribed Dirichlet and Neumann boundary conditions, respectively. These conditions either incorporate prescribed displacements along the part Γ D , namely u ( t ) = g ( t ) where t denotes time characterising changes in the applied load, or applied forces along the part Γ N which control surface tractions p by the relation p ( t ) = f ( t ), this case also includes a load free part of the boundary. The of such a deformable body is shown in Fig. 1.