PSI - Issue 42

Tamás Fekete et al. / Procedia Structural Integrity 42 (2022) 1467–1474 Tamás Fekete, Éva Feketéné-Szakos / Structural Integrity Procedia 00 (2019) 000–000

1471

5

0 ρ Ψ ε complemented by the constitutive equations: , , t s = − ∂ = ∂ = − ∂ Σ g ⌢ ⌢ ⌢ ɶ T i Ψ

⌢

ρ Ψ + ⌢ ⌢

(

)

G ɶ



0 ρ Ψ α ɶ ,

= −∂

K dV

A ɶ

j

i

j

V

t

The fields within these equations are dependent on ( ) , ˆ τ τ , except for the kinetic energy or kin E K ⌢ ⌢

, the thermal

th E ⌢ and the Helmholtz free energy

H E or Ψ ⌢ ⌢

energy

, all of which depend indirectly on time.

t d A stands for the time

t d = v u represents velocity in MCS . t V denotes the time-dependent volume,

t V ∂ means the

derivative of a field A ,

1 − = Σ F σ F stands for the t T j −

boundary of t V ; n is the outer normal to an elementary surface area, j =det( F ), second Piola-Kirchhoff stress tensor. ρ means mass density, σ denotes the stress tensor, ext f ⌢ of external forces, T is temperature, s ⌢ means entropy density, i s t

stands for the density

⌢ is the irreversible part of the entropy density, q j i α ɶ ( i = 1 … m ) are the internal variables describing short-length

denotes heat flux,

s j represents the entropy flux,

i g ɶ ( i = 1 … m ) are the thermodynamic driving

processes in the bulk responsible for dissipative reconfigurations and

forces conjugated to them. k A ɶ ( k = 1 … n ) stand for the internal variables representing macroscopic crack k G ɶ is the thermodynamic driving force –the generalized energy release rate–, which is conjugated to the k -th crack variable. The bulk dissipation rate produced by the reconfiguration processes is i t i d ⋅ g α ɶ ɶ (the Einstein summation convention is used). The dissipation rate generated by the crack propagation is k t k G d A ⋅ ɶ ɶ . The behavior of a crack-front –propagating with V c in MCS at ( ) , ˆ τ τ – in domain V with a surface area A is given by the generalized ( ) ( ) ˆ ... and ... J J integrals as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 1 ˆ ˆ ˆ ˆ ˆ ..., , , ..., , lim ..., , ..., , 1 ˆ ˆ ˆ ..., , ..., , ( ) t c A A A A t ext i i A i V V V V J K dA J J G A J J K dV dV sT dV j dV A τ τ ρ Ψ τ τ τ τ τ τ τ τ τ τ Ψ ρ ρ → − = + ⋅ + ⋅ = =   = − ∂ + − ⋅ + + ⋅           V σ v n f v g α ⌢ ⌢ ɶ ɺ ɺ ⌢ ⌢ ⌢ ⌢ ɺ ɶ ɺ ɺ (6) Equations (4) – (6) describe the Thermomechanical behavior of a solid body, including cracks. The description can be divided into three relatively independent, but highly entangled parts. Eq. (4) is a second-order kinematic model of a classic solid Boltzmann continuum. Eq. (5) describes the volumetric aspects of the processes under consideration using a complete system of mass, momentum, angular momentum, energy, and entropy balances, resulting in a unified Mechanical-Energetic-Entropic approach. This theory goes beyond CTIP by incorporating both the presence of dynamic dissipative processes in the body through the i α ɶ local internal variables, and the influence of cracks through the k A ɶ variables into the description. Comparing Eq. (5.b), (5.d) and (5.e), it can be concluded that the motion of the body is influenced by the energetic and dissipative processes, and vice versa. It is clear from a closer examination of Eq. (6) that in NLFTFM , crack-front behavior is described by more sophisticated time dependent expressions –the J and the ˆ J integral– than in the classic J -integral. J and ˆ J are thermodynamically consistent generalizations of the original J , extended to dynamic crack propagation. Crack-front stability is described by a time-dependent criterion ( ) ( ) ˆ ˆ ..., , ..., , G R τ τ τ τ = ɶ ɶ , a form like the classic J I = J Ic . The path independence hypothesis has been dropped from the model, and the sum of the Helmholtz free energy and kinetic energy density is in the core of the integral replacing the strain energy density, which indicates the dynamic nature of the description. When the effect of other fields on fracture is being considered, the Helmholtz Potential ( HP ) may be replaced by another suitable thermodynamic potential. As dictated by the 2 nd law of Thermodynamics, through dissipation, the ageing processes continually reduce the HP ; the more the free energy –available for deformation– decreases, the more the material becomes brittle. Note that the theory of Material Forces provides a similar framework to NLFTFM –see Steinmann (2022)–. propagation, and each

3.2. An Ageing Model

It was noted earlier that the NLFTFM framework is expected to be used for SIC s of LSS s with LTO . The framework presented in Eqs. (4) – (6) illustrates that it contains nonlinearities, thus its stability characteristics may be different from what is expected based on prior experience in linear systems studies. Therefore, qualitative theoretical assessments are valuable here.