PSI - Issue 48

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

305

4

 1 2   C I ε defines the Green-Lagrange strain tensor 

T  C F F .

The right Cauchy-Green tensor is specified by

– see e.g., Béda, Kozák, Verhás (1995) – . Picking the displacement vector u as primary variable, X X    H u u – the displacement gradient – enters into the picture, and is linked to F through   H F I . The Green-Lagrange tensor then acquires the following form – see e.g., Bažant, Cedolin (1991), Béda, Kozák, Verhás (1995), – :       1 2 , , T T         H H H H ε (4) The system behavior is governed by a combined set of the motion equations together with the field equations. The field equations can be subdivided into general balances valid for all material models and constitutive equations governing the behavior of the individual materials –see e.g., Béda, Kozák, Verhás (1995) , Steinmann (2022) – . For NLFTFM – see Chen, Mai (2013) – , the field equations are divided into two further categories: the first category includes equations describing volumetric phenomena and the second category includes equations describing the surface phenomena around cracks. The general balances and the defining equations for energy densities are:

           

t    t

0

the mass balance

d dV 



t

V

V 



thelinearmomentum balance

d dV  v

σ n

f

dA

dV

 





t

ext

V

V



t

t













V 



theangularmomentum balance

dV     r v r σ n 

 r f

d

dA

dV

 

t

ext

V

V



t

t

t

st (1 law of thermodynamics)

theenergy balance

























: σ v

dV       j σ f 

v

H th E E dV

dV

dV

d E 

 





X q

X

ext

X

t

kin

V

V

V

V

t

t

t

t

(5)

  

nd (2 law of thermodynamics)

thedissipation inequality



















V 





j

T dA j n

j

T d s dV 

T dV sdT dV 

  

  



t i

q

s

s

X

t

V

V

V



t

t

t

t





ε







1

t

0

:



d dV j    v v

Σ

H E dV

d dV d E  







1 2

t

t

t

kin

V

V

V

t

t

t

with:





        

 ; , , E K A E     ε kin i j H

thekineticenergy density

v α X



, , T T A X i

, , ,

theHelmholtz free energydensity thethermal energy density

α X

j

E T s  

th

that, are supplemented by the following constitutive equations:







t

,  

,

,  

Σ

g

s

G

K dV 

 





 



 

(6)

ε

0

0

α

T

i

j

A

i

j

V

t

As mentioned earlier, the fields within equations (5) and (6) are explicitly dependent on   ,   , except for the kinetic energy density ( or kin E K ), the thermal energy density ( th E ) and the Helmholtz Free Energy Density ( H E or  , HFED ), which depend implicitly on the time. t d A is the time derivative of a field A , t d  v u is velocity in MCS . t V means the time-dependent volume, t V  denotes boundary of t V ; n represents the outer normal to an elementary surface area, j =det( F ), 1 t t T j    Σ F σ F denotes the second Piola-Kirchhoff stress tensor.  indicates mass density, σ represents the stress tensor, ext f is the density of external forces, T denotes temperature, s represents entropy density, i s is the irreversible part of the entropy density, q j stands for the heat flux, s j means the entropy flux, i α ( i = 1 … m ) denote the internal variables representing short length-scale processes in the bulk, being responsible for dissipative reconfigurations, together with i g ( i = 1 … m ) as their conjugate thermodynamic driving forces.. k A ( k = 1 … n ) are the internal variables describing macroscopic cracks, and k G represent the thermodynamic driving force – the generalized