PSI - Issue 43

Vladislav Kozák et al. / Procedia Structural Integrity 43 (2023) 47–52

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V. Koza´k & J. Vala / Structural Integrity Procedia 00 (2023) 000–000

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tion. The dot here indicates the partial derivative according to time, F is a linear functional and A ( . ) and G ( . ) are (rather special) mappings defined on V ; our goal is to make the abstract function u , showing a certain time interval, satisfied the integral equation at each time t , Eq. (1) and, in addition, complied with certain prescribed (especially marginal and initial) conditions. The main di ffi culty of the hypothetical direct approach is related to the nonlinearity of A ( . ), thus certain A ∗ ( . , . ) is considered as an approximation of A ( . ), to force the linearity of A ∗ ( . , . ) in its 1st argument, open to the later iterative improvement related to its 2nd argument. Thus Eq. (1) can be rewritten in its time-discretized form G ( u s − u s − 1 ) , v + h A ∗ ( u s − 1 , u s ) , v = F s , v , (2) where u s and u s − 1 approximate the unknown abstract function u in discrete times t = sh a t = ( s − 1) h , same as F s in case of unknown F ; here s ∈ { 1 , ..., m } for m = τ/ h , where τ is the length of the considered time interval (0 ≤ t ≤ τ ), where we need the limit case h → 0, so m → ∞ . After adding some additional conditions and using the standard norm . in the space V , from Eq. (2) we get u s 2 + s r = 1 u r − u r − 1 2 ≤ c   u 0 2 + h s r = 1 F r 2   , (3) where c is a generic constant. Thanks to the convergence properties of (in the simplest case) 3 types of Rothe se quences: i) the linear Lagrange splines u m ( t ) = u s − 1 + (( t − ( s − 1) h ) / h )( u s − u s − 1 ), ii) the standard simple functions ¯ u m ( t ) = u s and iii) the retarded simple functions ˘ u m ( t ) = u s − 1 , assuming ( s − 1) h < t ≤ sh and s ∈ { 1 , . . . , m } , taking the a priori estimates of the type (3) into account, u can be seen as certain weak limit in { u m } ∞ m = 1 in a reasonable sense. This approach is open to various generalizations, including the fully dynamic formulation, as discussed in (Vala and Koza´k, to be published); however, in this short paper we shall confine ourselves to the computational approaches compatible with Eq. (2).

3. Approximate solutions using XFEM

Fibre cement composites belong to the class of promising concretes with higher mechanical crack resistance. This allows for a finer and more economical design; is therefore a new perspective on the creation of building structures or the replacement of steel structures is necessary. These structures exposed to loads can result in stresses in the body exceeding strength of the material, thus leading to gradual failure. The XFEM can be used for practical calculations

Fig. 2. Crack propagation using XFEM, HE is Heaviside Enrichment, CTE Crack Tip Enrichment.

and the extended finite element method (XFEM), see Fig. 2, working with adaptive enrichment of a set of basis functions near singularities (Bouhala et al., 2013). Fig. 2 schematically illustrates the following Eq. (4) and shows how the crack propagation inside the element (second term) and in front of the crack front (third term) is modelled. This method (including a number of its modifications with its own names and designations) already has a relatively rich history; progress in recent years can be seen by comparing the founding works (Hillerborg et al., 1976) and (Havla´sek et al., 2016) with (Pike and Oskay, 2015) and (Sorensen and Jacobsen, 1998). More general procedure for randomly oriented fibres presents (Brighenti, 2012).