PSI - Issue 46

Tamás Fekete et al. / Procedia Structural Integrity 46 (2023) 189–196 Tamás Fekete / Structural Integrity Procedia 00 (2021) 000–000

194 6

 X F χ is deformation gradient, j =det( F ),

of displacements,

T  C F F is right Cauchy-Green tensor,





t d  v u means

1 2   E C I is the Green-Lagrangian strain tensor. t d  A A

 is Lagrange time derivative of a field,

1

t

t

T

velocity.   Σ F σ F is the second Piola-Kirchhoff stress tensor. Based on the ideas of Chen and Mai (2013), the governing field equations of the model –i.e., the balances and the constitutive laws– are as follows: j 

t    t

0

mass balance

d dV 



t

V

f 

V 



linearmomentum balance

d dV  v

σ n

dA

dV

 





t

ext

V

V



t

t















V 



angularmomentum balance

dV     r v r σ n 

 r f

d

dA

dV

 

t

ext

V

V



t

t

t

st energy balance (1 law of thermodynamics)



 H th d E E E dV        t kin





















: σ v

dV      j σ f 

v

dV

dV







X q

X

ext

X

V

V

V

V

t

t

t

t

nd dissipation requirement (2 law of thermodynamics)























V 





(1)

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

 











1

t

:

0



v v

Σ

d dV d E E dV    C

dV j 

   



1 2

t

t

kin

H

V

V

V

t

t

t

with:



E  E 





; , , 

kineticenergy density

K A v α

X



kin

i

j

 





, , T T A  α  X i

, , ,

Helmholtz free energydensity thermal energy density

C

X



H

j





E T s  

th

as well as the constitutive relations: 2 , , t s       Σ   



 K      



G 

,  

g 

dV

 



0

0

C

α 

T

i

j

A

i

j

t V

In these equations, all physical fields and other variables are explicitly time-dependent, except for kinetic, thermal and Helmholtz free energy, which are time-dependent implicitly, through their variables;  means mass density, σ denotes stress, ext f  is density of external forces, T means temperature, s  is entropy density, i s  is the irreversible part of the entropy density, q j denotes heat flux, s j represent entropy flux, n is outer normal to a surface area, t V denotes time-dependent volume, t V  represents boundary of t V ; i α ( i = 1 … m ) symbolize inner variables describing short length-scale processes in bulk resulting in dissipative reconfigurations, i g  ( i = 1 … m ) the thermodynamic driving forces conjugated to them, while k A ( k = 1 … n ) denote the inner variables representing macroscopic crack propaga tion, and each k G  is the thermodynamic driving force –called generalized energy release rate– conjugated to the k -th crack variable. The bulk dissipation rate caused by the reconfiguration processes is i i  g α   , and the dissipation rate accompanying crack propagation is k k G A    , –where the Einstein summation convention is used–. For a domain of volume V , with surface area A , containing a crack with propagation velocity V c in the reference frame, crack front behaviour at  is described by the following generalised     ˆ ..., and ..., J J   integrals:

1





 K     

ˆ

















A 

G 

t V σ v n

...,

,

...,

lim ..., J

...,

J

dA

J



 

   











c

A 

A

A

0

A



(2)

  

  

1



K dV  





 

 ˆ ..., J

















1

...,

(

)

ext f v 

 g α  



J

dV sTdV  

j

dV



  

  







t

i

i

A 

A

i

V

V

V

V