PSI - Issue 43

Vladislav Kozák et al. / Procedia Structural Integrity 43 (2023) 47–52 V. Koza´k & J. Vala / Structural Integrity Procedia 00 (2023) 000–000

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4. Modified extended finite element technique

In XFEM, the mesh is independent of the internal boundaries such as material interfaces and cracks. These internal boundaries usually cause weak or strong field discontinuities variables that will be taken into account in XFEM by incorporating enrichment functions into the standard FEM approximation. Within XFEM, following (Vilppo et al., 2021), (Eringen, 1984), (Eringen, 2002) and (Evgrafov and Bellido, 2019), in an arbitrary time step t = sh , s ∈ { 1 , . . . , m } , the approximation of a displacement field u (related to the geometric configuration for t = 0 as a reference one) in the element for a viscoelastic body with a crack can be expressed as u XFEM ( x ) = i ∈ C A N i ( x ) u i + j ∈ C S N j ( x ) H j ( x ) a j + k ∈ C T N k ( x ) 4 n = 1 Φ n k ( x ) c n k ; (4) here C A , C S , C T are sets of points corresponding to Fig. 3 (left part), N i ( x ) are standard nodal shape functions, H j ( x ) are Heaviside functions, needed at crack interfaces, and Φ n k ( x ) are additional shape functions, designed to handle the expected phenomena at crack tips. For the simplified plane stress formulations Φ n k ( x ) are typically expressed in polar coordinates ( θ, ρ ), i. e. x 1 = ρ cos θ , x 2 = ρ sin θ , for n ∈ { 1 , 2 , 3 , 4 } in the form Φ k ( x ) = [ √ ρ sin( θ/ 2) , √ ρ cos( θ/ 2) , √ ρ sin( θ/ 2) sin θ, √ ρ cos( θ/ 2) sin θ ] . (5) The computational formula Eq. (5) is open to natural generalizations, working with more elements of C T , adaptable both to classical (extrinsic) and intrinsic XFEM techniques. Thus, beyond the standard degrees of freedom (DOFs) u i , corresponding to C A (standard FEM without crack growth), one must deal with certain additional DOFs a i and c i , related to C S (enrichment nodes) and C T (crack tip enrichment nodes) adaptively. In general, the first term of Eq. (4) corresponds to the standard finite element method, the second realizes the crack formation and the third the criterion of formation, whereby Φ m k represents the local situation close to the crack front. On Fig. 3 (right part) one can see 5

Fig. 3. Stress determination ahead the crack tip using nonlocal approach.

types of finite elements: A refers to usual ones, B to neighbours to elements with cracks, C to crack tip elements, D to boundary elements neighbour to cracks and E to elements neighbour to C. Fig. 3 (left part) shows the procedure for calculating stress concentration according to (Eringen, 1984).

5. Smeared damage implementation

Since the proper quantitative analysis of formation and propagation of such complicated crack systems is not easy, particular attention is devoted to the Eringen’s model for generating the multiplicative damage factor, to obtain an initial- and boundary-value problem for certain parabolic system of partial di ff erential equations (in the quasi-static case), or even for the hyperbolic one (in the fully dynamic case), including the design of appropriate computational