PSI - Issue 74

Jiří Vala et al. / Procedia Structural Integrity 74 (2025) 91–98

93

J.Vala & V.Koza´k / Structural Integrity Procedia 00 (2025) 000–000

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finite strain analysis, corresponding to such physical formulations, then needs the theory of structured deformation by Kruzˇ´ık and Roub´ıcˇek (2019), Chapters 8 and 9, and Kro¨mer et al. (2024). 2. Physical and mathematical preliminaries Let us consider a deformable body Ω , occupying a domain in the 3-dimensional Euclidean space R 3 , consisting of a finite number n of disjoint parts Ω i with i ∈ { 1 , . . . , n } with Lipschitzian boundaries Λ i j between Ω i and Ω j where i , j ∈ { 1 , . . . , n } , j i (some of them are usually empty). Such sets generate both the exterior boundary ∂ Ω of Ω and the set of all interior interfaces Λ , whose properties should be introduced separately. Standard boundary conditions can be implemented on ∂ Ω , namely ∂ Ω can be divided into two disjoint parts: Θ with Dirichlet boundary conditions (this must have a non-zero Hausdorf measure on ∂ Ω , to avoid unsu ffi cient support in the following considerations) and Γ with Neumann boundary conditions. A Cartesian coordinate system x = ( x 1 , x 2 , x 3 ) can be considered in R 3 , together with the displacement u ( x , t ) = ( u 1 ( x , t ) , u 2 ( x , t ) , u 3 ( x , t )) of any point of Ω , Θ , Γ and Λ , dependent on time t , related to its initial state with t = 0. Thus, for a time interval I = [0 , T ] of a finite length T , we are able to set the Cauchy initial condition u ( ., 0) = o on Ω where o = (0 , 0 , 0). The following Lebesgue and Sobolev function spaces, as introduced by Roub´ıcˇek (2005), Part 1.2, will be needed: H = L 2 ( Ω ) 3 , E = L 2 ( Ω ) 3 × 3 sym , G = L 2 ( Γ ) 3 , G = L 2 ( Λ ) 3 , Z = L ∞ ( Ω ), M = L 2 ( Ω ) (3 × 3) × (3 × 3) sym and V = { v ∈ W 1 , 2 ( Ω ) 3 : v = o on Θ } . The brief notation of scalar products in special Hilbert spaces will be used: ( . , . ) in H , ⟨ . , . ⟩ in G⟨ . , . ⟩ ∗ in G , ( . , . ) in E . For a fixed t ∈I and an arbitrary γ ∈E , then [( γ,τ ) ] must be understand in the sense of Bochner integral of an abstract function (( γ,τ ( . , ˜ t )) over ˜ t ∈ [0 , t ] where τ ∈ L 2 ( I , E ), see Roub´ıcˇek (2005), Part 7.1. To simplify our formulations as much as possible, in a model problem we shall work with the linear strain tensor ε ( v ) ∈ E with components ε i j ( v ) = ( ∂ v i /∂ x j + ∂ v j /∂ x i ) / 2where i , j ∈ { 1 , 2 , 3 } for any virtual displacement v ∈V . Some potential generalizations (with much less details) will be sketched later. From the physical point of view, the principles of classical thermodynamics can be exploited, namely of conserva tion of mass, (linear and angular) momentum and energy. Using the Boltzmann continuum theory, we shall see that a model problem can be formulated in its weak (integral) form from the conservation of momentum only, supplied by appropriate constitutive equations between a) the total stress σ ∈ L 2 ( I , E ) and the strain ε ( u ) on Ω and b) the interior surface traction ς ∈ L 2 ( I , G ) and the interface jumps [ u ] of u (in the sense of traces), whose orientation must be pre-defined on all parts of Λ .

stress τ α C

β C

stress σ − τ (1 −D ) C

Fig. 1. Scheme of the viscoelastic SLS model with damage.

For the case a) the generalized Kelvin (parallel) and Maxwell (serial) viscoelastic chains, consisting of elastic and viscous components, by Trcala et al. (2024a) (without damage) and Trcala et al. (2024b) (with smeared damage) can be implemented; for more general compositions of such chains cf. Al Janaideh (2024). Here we shall work, for simplicity, with the so-called standard linear solid (SLS) model only, presented by Fig. 1. Here C∈M is the sti ff ness tensor, α,β ∈ Z are certain scalar material factors and D ∈ Z is considered as a damage factor, whose increase forces the loss of sti ff ness. Remind that for an isotropic material C contains only 2 independent characteristics, well known as the Lame´ constants (which can be variable on Ω ), or, in the alternative form, the Young modulus and the Poisson constant. In particular, for α = β = D = 0 everywhere Fig. 1 represents the Hooke purely elastic model, for α = D = 0 and a positive β the classical Kelvin viscoelastic model, etc. It is natural to assume that there exists such positive constant c thatmax( α ( x ) ,β ( x )) ≥ c holds for any x ∈ Ω together with A T C ( x ) A ≥ c A T A for an arbitrary matrix A ∈ R 3 × 3 sym . Then σ − τ,τ ∈ L 2 ( I , E ) in particular branches of Fig. 1 can be seen as two components of the total stress σ ∈ L 2 ( I , E ).