PSI - Issue 48

Tamás Fekete / Procedia Structural Integrity 48 (2023) 302–309 Fekete / Structural Integrity Procedia 00 (2023) 000 – 000

306

5

energy release rate – , conjugated to the k -th crack variable. Rate of dissipation produced by bulk reconfiguration processes is i t i d  g α (the Einstein summation convention is used). The rate of dissipation generated by crack propagation is calculated by k t k G d A  . Crack front behavior of a crack propagating at velocity V c in MCS at   ,   , in the domain V enclosed by the surface A , is described by the generalized     ˆ ... and ... J J integrals according to:             1 ˆ ..., , , ..., , lim ..., , ..., , T c J K dA J J G                  V σ v n

A 

A

A

A

0

A



(7)

  

  

1





ˆ



















1

..., ,

..., ,

(

)



ext f v 

 g α

J

J

K dV

dV sTdV  

j

dV

 

 

  



  



t

i

i

A

A

i

V

V

V

V

Equations (4), (5), (6) and (7) respectively represent the thermomechanical behavior of a solid body including macroscopic cracks. The model can be partitioned into three relatively separate but strongly entangled parts. Equation (4) presents a second-order kinematic model of a classical solid Boltzmann continuum. Field equations (5) and (6) describe the processes under investigation with a complete set of Mass, Momentum, Moment of Momentum, Energy and Entropy balance equations, implying that the description is based on a unified Mechanical-Energy-Entropy approach. The theory goes beyond the approach taken by the original CTIP , as it is a Thermodynamics with Internal Variables ( TIV ) with local internal variables. The i α internal variables account for the dynamic dissipative processes in the body acting at microscopic or even submicroscopic length-scales, while the k A internal variables incorporate the behavior of macroscopic cracks. By comparing equations (5. α ), (5. δ ) and (5. ε ), the conclusion may be drawn that on the one hand, the energetic and dissipative processes influence the motions of the body, and on the other hand, the motions also affect the energetic and the dissipative processes. Looking more closely at equation (7), it becomes clear that the crack front behavior in NLFTFM is described in much more sophisticated time-dependent terms – the J and the ˆ J integral – than the J -integral developed by Rice – Rice (1968) – , which can be written in the following form:   T e A J WI dA          u σ n (8) As is known, Rice ’s J -integral is integration path independent. This independence does not, however, stem from deep physical principles, but from the simplifying assumption that Rice's studies were for globally homogeneous elastic materials. For homogeneous materials, the path independence of J follows from Green's theorem, well known from analysis. For inhomogeneous materials, the situation is more sophisticated. This is why J and the ˆ J appear in equation (7). ˆ J and J is a generalized J -integral, based entirely on Thermomechanics. ˆ J is the Crack-Tip Driving Force ( CTDF ) in inhomogeneous materials , which is also suitable for investigating dynamic fracture phenomena . The use of the HFED in the core of the integral paves the way towards a multifield description of fracture mechanics problems. If the effect of other fields on fracture is intended to be investigated, the HFED may be replaced by other adequate thermodynamic potential. In this theory, crack front stability is described by a time-dependent,     ..., , ..., , G R      criterion, which appears to be similar to the classic J I = J Ic , but is obviously much more refined. Note that the theory of Material Forces provides a similar framework to NLFTFM – see Steinmann (2022) – . To sum up, the sketched, Modern Thermomechanics based NLFTFM is a robust and reliable theory framework. Its integral part is the thermodynamical ˆ J -integral, defined for inhomogeneous materials. The emergence of NLFTFM represents a major paradigm shift in FM (see Fig. 3.). The approach is incorporated into the new SIC methodology.

Fig. 3. Paradigm shift in FM . In NLFTFM , the FM of inhomogeneous materials is an obvious implication.