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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The evaluation of D is rather delicate, due to its dependence of some nonlocal version of vectors of invariants derived from ε ( u ), σ , or similar matrix quantities. Let us assume zero values of D on Ω for t = 0. The following sequential five-step algorithm, motivated by Vala (2023), shows such an evaluation of D for any fixed positive t ∈I : 1. Find three scalar principal values ϵ i ( u ) with i ∈ { 1 , 2 , 3 } , corresponding to ε ( u ): det ( ε ( u ) − ϵ i ( u ) I ) = 0, I ∈ R 3 × 3 being the unit matrix. 2. Calculate an equivalent strain ˜ ε ( u ) = ω ( − ϵ 1 − ( u ) , ϵ 1 + ( u ) , − ϵ 2 − ( u ) , ϵ 2 + ( u ) , − ϵ 3 − ( u ) , ϵ 3 + ( u )); ϵ i + ( u ) , ϵ i − ( u ) with i ∈ { 1 , 2 , 3 } are the positive and negative parts of ϵ i , a bounded continuous function ω of 6 real non-negative argu ments must be prescribed. 3. Convert ˜ ε ( u ) to its nonlocal form ¯ ε ( u ) by Eringen (2002), i. e. ¯ ε ( u ( x , t )) = Ω K ( x , ˜ x ) ˜ ε ( u (˜ x , t ))d˜ x , Ω K ( x , ˜ x )d˜ x = 1 for all x ∈ Ω , using a regularizing kernel K ∈ L 2 ( Ω × Ω ). 4. Evaluate a trial value of damage factor ˆ D ( u ) = ϖ (¯ ε ( u )); a non-decreasing real continuous function ϖ must be prescribed. 5. Force the irreversibility of damage, taking all times ˜ t ∈ [0 , t ] into account, as D ( u ( ., t )) = max 0 ≤ ˜ t ≤ t ˆ D ( u ( ., ˜ t )). For the case b) the crucial step is the evaluation of ς = T ([ u ]) working with a bounded continuous function T of three variables, related to 1 normal and 2 tangential directions on all parts of Λ . Such evaluation, under the assumption of cohesive interfaces, discussed in Pike and Oskay (2015), Koza´k et al. (2017) and Liu at al. (2022), is usually designed for crack tips, their open and closed parts, etc., specifically, respecting three basic traditionally distinguished modes: I opening, II shearing, and III tearing, due to the orientation of forces relative to the crack. For both cases an appropriate setting of all presented material parameters and functions from well-organized laboratory and / or in situ experiments is expected. 3. A model problem and its solvability Te present a model problem in the Boltzmann continuum theory, we can start with the weak form of the conserva tion of momentum ( ε ( v ) ,σ ) + ⟨ [ v ] ,ς ⟩ ∗ = ( v , f ) + ⟨ v , g ⟩ (1) for an arbitrary v ∈ V and any t ∈ I . Whereas the evaluation of ς comes from the preceding discussion directly, the evaluation of σ must take both branches of the scheme on Fig. 1 into consideration: whereas the lower one gives σ = τ + (1 −D ( u )) C ε ( u ) directly, the upper one takes the form of an additional integral equation with an unknown τ ( γ, C − 1 τ/α ) + [( γ, C − 1 τ/β )] = ( γ,ε ( u )) , (2) whereas (1) takes the form ( ε ( v ) ,τ ) + ( ε ( v ) , (1 −D ( u m s )) C ε ( u )) + ⟨ [ v ] , T ( u ) ⟩ ∗ = ( v , f ) + ⟨ v , g ⟩ . (3) To avoid technical di ffi culties in proofs, we are allowed to assume f ∈ L 2 ( I , H ) for volume loads and g ∈ W 1 , 2 ( I , G ) for surface loads in (1). Then a model problem can be read as: find such u ∈ L 2 ( I , V ) and τ ∈ L 2 ( I , E ) that (3) and (2) hold for any t ∈I ; two initial conditions u = o and τ = O are considered, O being the zero matrix from R 3 × 3 . Since the unknowns u and τ (3) and (2) belong to infinite-dimensional abstract function spaces and their (semi- )analytical form is usually not available, we need to implement some discretisation technique. We can come out from the method of the discretisation in time, following Roub´ıcˇek (2005), Part 8.1: I is covered by a finite number m of subintervals I m s , defined as the sets of all t satisfying ( s − 1) h < t ≤ sh where s ∈ { 1 , . . . , m } . The notation h = T / m will be also used for the time step for brevity; the analysis of limit passage m →∞ (or h → 0 + ) is necessary.. The time-discretised formulation of (3), motivated by the classical Euler implicit formula (notice the nonlinear functions in the second and third left-hand-side additive terms), is ( ε ( v ) ,τ m s ) + ( ε ( v ) , (1 −D ( u m s )) C ε ( u m s )) + ⟨ [ v ] , T ( u m s ) ⟩ ∗ = ( v , f m s ) + ⟨ v , g m s ⟩ (4) for any s ∈ { 1 , . . . , m } . Notice that, for arbitrary fixed s and m , f m s ∈H and g m s ∈ G can be considered as su ffi ciently good approximations of f and g on I m s , e.g., as Cle´ment quasi-interpolations by Roub´ıcˇek (2005), Part 8.2, based on the generalised mean values on such abstract functions on I m s . Unlike this, u m s ∈V ,and τ m s ∈H refer to approximations of unknown values of u m s ∈V and τ m s ∈ E valid on I m s . Applying the rectangular rule for numerical quadrature, from (2) we analogously deduce ( γ, C − 1 τ m s /α ) + h ( γ, C − 1 ( τ m 1 + . . . + τ m s ) /β ) = ( γ,ε ( u m s )) . (5)