PSI - Issue 52

Roman Vodička et al. / Procedia Structural Integrity 52 (2024) 242 – 251 R. Vodicˇka / Structural Integrity Procedia 00 (2023) 000–000

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to volumetric or shear strain independently, and to separate tensile ( + ) and compressive (-) parts of the spherical tensor in order not to degrade under compression. Another ingredient to the energy balance is the kinetic energy which is expressed in a standard form K (˙ u ) = Ω 1 2 ρ | ˙ u | 2 d Ω , (3)

and energy of the external forces, here, represented by the boundary forces f F ( t ; u ) = Γ N f ( t ) · u d Γ .

(4)

In the material, there may appear processes which dissipate energy. One part of dissipation is caused be viscoelastic character of the material and pertinent integrals correspond to dampers in the scheme presented in Fig. 1. Second, if the crack tends to evolve in other than opening mode (Mode I, cf. using superscript I in the fracture energy), there may appear another nonlinearity which dissipates the energy. The nonlinearity can be mimicked by introducing the mode dependent fracture energy, see e.g. Benzeggagh and Kenane (1996). Additionally, the crack propagation is a unidirectional process, in terms of the internal parameter α it corresponds to a constraint ˙ α ≤ 0 in Ω . All these assumptions can be indicated by a dissipation pseudo-potential R ( u ,α ; ˙ u , ˙ e 2 , ˙ α ) = Ω Φ ( α ) τ r1 K p sph + e (˙ u ) 2 + τ r2 K p sph + ˙ e 2 2 + τ r1 µ | dev e (˙ u ) | 2 + τ r2 µ | dev ˙ e 2 | 2 + τ r1 K p sph − e (˙ u ) 2 + τ r2 K p sph − ˙ e 2 2 − 3 8 ε G c ( u , e 2 ) − G I c ˙ α η d Ω (5) provided that the constraining inequality for ˙ α is satisfied. The fracture mode dependence expressed by the function as G c ( u , e 2 ) takes a form which for an opening crack reduces to G I c (so that the last term vanishes and no additional energy is dissipated), and there remains just G II c in the case of the Mode II crack (shear), which, if desired, adds some dissipated energy if G II c − G I c is positive. A function which reflects such a requirement is e.g. The dynamic evolution of a deformable body with a general linear solid rheology and with cracks can be de duced from Hamilton variational principle with an extension for dissipative systems, see Bedford (1985); Kruzˇ´ık and Roub´ıcˇek (2019), which says that the action,i. e. the integral T 0 K (˙ u ) − E ( t ; u , e 2 ,α ) + F ( t ; u ) − Ω ∂ ˙ u R · u + ∂ ˙ e 2 R · e 2 + ∂ ˙ α R · α d Ω d t (7) over the fixed time interval [0,T] during which the system evolves, is stationary respectively to the trajectory ( u ( t ) , e 2 ( t ) ,α ( t )) with t ∈ [0 , T ]. The expression in the first parentheses introduces Lagrangian, and in the second parentheses the triple ∂ ˙ u R ,∂ ˙ e 2 R ,∂ ˙ α R defines nonconservative dissipative force in relation to dissipation functional from Eq. (5). The condition of stationarity provides the equations of motion and of flow rules for internal variables in the following form expressed as nonlinear inclusions with initial conditions G c ( u , e 2 ) = K p sph + ( e ( u ) − e 2 ) 2 + ( e ( u ) − e + γ sph + e 2 + γ | sph + e 2 ) | 2 2 2 2 | + µ | dev( e ( u ) − e 2 ) | 2 µ ( | dev( e ( u ) − e 2 ) | 2 + γ | dev e 2 | + γ | dev e 2 | 2 K p | sph G I c + 2 ) G II c . (6)

d d t K (˙ u ) + ∂ u E ( t ; u , e 2 ,α ) + ∂ ˙ u R ( u ,α ; ˙ u , ˙ e 2 , ˙ α ) F ( t ; u ) ,

u (0 , · ) = u 0 , ˙ u (0 , · ) = v 0 ,

0 − τ r1 e ( v 0 ) ,

γ 1 + γ C

(8)

− 1 σ

∂ e 2 E ( t ; u , e 2 ,α ) + ∂ ˙ e 2 R ( u ,α ; ˙ u , ˙ e 2 , ˙ α ) 0, ∂ α E ( t ; u , e 2 ,α ) + ∂ ˙ α R ( u ,α ; ˙ u , ˙ e 2 , ˙ α ) 0,

e 2 (0 , · ) = e 20 = e ( u 0 ) −

α (0 , · ) = α 0 = 1,