PSI - Issue 23

Jiří Vala et al. / Procedia Structural Integrity 23 (2019) 328 – 333 Ji ří Vala, Vladislav Kozák / Structural Integrity Procedia 00 (2019) 000 – 000

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A possible approach to simulate the propagation of cracks is the application of softening material formulations to continuum elements leading to a smeared representation of the crack path. An alternative discrete approach imple ments the cohesive finite elements, Ko zák & Chlup (2011), Koz ák et al (2017), Vala (2017) ; in this case the crack path must be known a priori, or all element interfaces have to be taken into account, which forces many new degrees of freedom, accompanied also by the risk of non-physical reduction of effective stiffness. The application of XFEM is able to suppress such drawbacks in the simulation of propagation of cohesive cracks; however, it must handle the non-existence of a sharp singularity at the crack tip, with more complicated derivation of required stresses from displacements. In general the complete computational model should involve the formation and propagation of cracks, their bridging by fibres, the loss of cohesion between fibres and matrix, their mutual sliding with friction and the fibre destruction; special functions are necessary e.g. for stress singularities in the case of crack opening and closing. The two-phase composite model of matrix and inclusion, based on the Eshelby solution and on the Mori Tanaka homogenization scheme, can be adapted to the directional propagation of microcracks, generalized also for long fibres by Bouhala et al (2013), Mihai & Jefferson (2017). Random spatial variability of material parameters can be handled using the stochastic simulation of damage, Eli áš et al (2015). The formulation of the related mathematical problem, incorporating reasonable physical and engineering simpli fications as the starting point for a derivation of the effective computational algorithm, comes from the principles of classical thermodynamics, namely from the 1st one of conservation of scalar quantities as energy, (linear and angu lar) momentum and mass and from the 2nd principle concerning the irreversibility of natural processes, here namely of the damage formation and propagation, which must be respected by semi-empirical (motivated from available information on material micro- and mesostructure) constitutive relationships, together with reasonable initial and boundary conditions. As with a simplified model example (useful just for this short paper), coming from such con siderations, we can start with the an abstract (in general nonlinear) quasi-static problem ( ), ( ), , G u v A u v F v   (1) where the brackets refer to some dualities for reflexive and separable Banach spaces V (in particular, to scalar prod ucts in Hilbert spaces) for any time t from some time interval [0, ] I   for a positive  (the limit case   is not forbidden, but not discussed here), v V  denotes a virtual quantity, e.g. the displacement related to the reference configuration, the dot means / t   , F is a linear functional and ( ) A  , ( ) G  are (rather special) mappings defined on V ; we are seeking for an ( ) u t mapping I to V coinciding with some prescribed 0 u V  for 0 t  (a Cauchy initial condition) formally. However, our real aim is to find u V  satisfying (1) with the vanished 1st additive term; the main difficulty of the hypothetical direct approach comes from the nonlinearity of ( ) A  . Whereas ( ) G  can be taken ( ) G  , an coercive operator ( ) A  may be useful to be decomposed as * ( , ) A   , linear in the 2nd variable and compact (if possible) in the 1st one, to apply the existence theory for weakly continuous (or pseudomonotone) operators. Consequently (1) can be transferred (accounting for the convergence properties of Rothe sequences) to 2. Physical, mathematical and computational background additional assumption we are allowed to come from (2) to the estimate, working with the norm  in V , 2 2 2 2 1 0 1 1 s s s r r r r r u u u c u h F                1 m m u u   , etc., with m  to zero. In particular, following Pike & Oskay (2005) (with several straightforward generalizations, working with geo metrical and physical linearization of elastic matrix and fibre behavior and with a macroscopic view to matrix crack ing, let us consider an open set (typically a domain)  in the 3-dimensional Euclidean space 3 R with an external boundary  , decomposed to 2 disjoint parts  (for homogeneous Dirichlet boundary conditions) and  (for non- (3) where c is some generic constant; (3) then manifests the decrease of 1 s s G u u v h A u u v h F v      1 s u  approximating an unknown u for t sh  and * 1 ( ), ( , ), , s s s (2) with s u , ( 1) t s h   , as well as s F in the case of known F (with certain final value); here {1,..., } s m  with / m h   , considering the limit passage 0 h  (thus m  ).Under some