PSI - Issue 74

Jiří Vala et al. / Procedia Structural Integrity 74 (2025) 91–98 J.Vala & V.Koza´k / Structural Integrity Procedia 00 (2025) 000–000

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fractured zones relies on some type of nonlocal reqularisation, as that by Eringen (2002); in particular, for cementitious composites a sequence of upgrades of the so-called Mazars model is available, cf. Giry et al. (2011), Havla´sek et al. (2016) and Arruda et al. (2023). The proper multiscale approach, compatible with Steinhauser (2008), similar to that of Han at al. (2017) relying on the properties of grain boundaries, supported by selected probabilistic considerations, su ff ers from incomplete and / or very large and complicated data for material descriptions, whose potential identifi cation from laboratory experiments generates as an additional serious problem, and forces large and time-consuming computations typically. However, a remarkable progress can be seen in the nonlocal macro-meso-scale coupling in the last years, cf. Lu et al. (2024). From the point of view of mathematical analysis, most above mentioned engineering formulations and related algo rithms refer to practical validation on special experimental datasets and need proper verification: existence of solution in some reasonable sense, namely in appropriate infinite-dimensional spaces of integrable (abstract) functions, con vergence of computational algorithms in finite-dimensional spaces, etc. Notice that this is seems to be a continuous process in the history of numerical mathematics: e. g. the first significant engineering results for the 2-dimensional stationary problems in continuum mechanics obtained from the finite element method come from the early 50ties of the 20th century, whereas the first formal mathematical existence and convergence proofs using the theory of Sobolev spaces occurred about 15 years later. Up to now, most such mathematical proofs are still related to certain model problems for partial di ff erential equations, supplied with appropriate boundary and initial conditions whose analysis can be performed exploiting available results from the theory of linear function spaces. All following considerations in this short conference paper can be seen as such compromise, too: we shall apply numerous results from Roub´ıcˇek (2005) and Dra´bek and Milota (2013) and leave most considerations beyond their scope to potential generalizations; they are not straightforward and contain unclosed additional problems typically. The former authors’ analyses of such type, whose partial results can be exploited here, can be found in Vala and Koza´k (2020) (quasi-static formulation, classical Kelvin viscoelastic model), Vala and Koza´k (2021) (partially dynamic formulation, classical Kelvin vis coelastic model), Vala (2023) (fully dynamic formulation, classical Kelvin viscoelastic model) and Vala and Toma´sˇ (2025) (quasi-static formulation, SLS viscoelastic model – see lower, smeared damage only). After this Introduction (Section 1), we shall continue with Physical and mathematical preliminaries (Section 2), which will enables us to discuss A model problem and its solvability (Section 3). Then we shall come to Computational issues (Section 4), containing an illustrative example, and finish with brief Conclusions (Section 5), including some research priorities for the near future. A model problem will be formulated in a rather simple way, to preserve the transparency from the point of view of (quasi-)linear functional analysis, but open to generalizations in various directions. It can be namely adopted for various classes of dynamic problems, which brings the transform from parabolic to hyperbolic problems mathemati cally, by Vala and Koza´k (2021) and Vala (2023). However, this does not cover fast dynamic processes by Neˇmec et al. (2022) with contacts / impacts of deformable bodies with potential cracking where the precise formulation of the dissipation energy on contacts seems to be most important, thus further generalization is needed. Unfortunately, such processes can be rarely analysed inside the small strain theory in a way acceptable for engineering practice; switching to the finite strain theory and geometrical updates like Le´ger and Pepin (2016) then make all proofs much more com plicated. Even our illustrative example will require a slight generalization in comparison with the linear viscoelastic scheme presented by Fig. 1. Another limitation of the above discussed approach lies in the a priori prescribed system of interfaces Λ , whereas some recently improved XFEM algorithms try to create a potential crack in an arbitrary direction. Unfortunately, the mathematical analysis of such algorithms and their limit behaviour requires deep results on function spaces introduced on non-Lipschitzian domains, like the partial ones from Jiang and Kauranen (2015) and Cianchi and Maz’ya (2016), where still unclosed questions can be expected. A great research challenge of last years is the possibility to come from the simplified formulations of selected con servation principles from classical thermodynamics, supplied by some (semi-)empiric constitutive equations, to the proper physical analysis, coming from the specific Gibbs energy and the complementary form of the dissipation po tential; a certain time-variable symmetric matrix of order 3 for any point of a deformable body can be then introduced as a measure of material damaging. Vilppo et al. (2021) expresses the specific Gibbs energy as a function of four state variables, namely temperature, stress, damage and a set of internal parameters, corresponding to four dissipative vari ables, involved in a dissipative potential, whose partial di ff erentiation works with the theory of subdi ff erentials. The