PSI - Issue 37

Fekete, Tamás et al. / Procedia Structural Integrity 37 (2022) 779–787 Fekete, T.: The Fundaments of Structural Integrity … / Structural Integrity Procedia 00 (2021) 000 – 000

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a framework theory addressing methodological issues of SI problems to LSPSs that will provide a coherent and comprehensive description of ageing and fracture phenomena. In the next, the fundamental relationships of such a thermodynamics-based model will be outlined in a nutshell. 3.3. Sketch of a thermodynamics-based model for Structural Integrity Calculations of Large-Scale Pressure Systems As shown above, modern thermodynamics is a sound scientific umbrella theory for the methodology of SICs for LSPSs . To construct methodology from the general theory for specific engineering problems, the key engineering problems must be identified, and then a problem-specific, yet general, theoretical model tailored to the problem area must be constructed. A cornerstone in SI of LSPSs is the identification of a basic model able to describe ageing and fracture mechanics simultaneously. In 1921, Griffith published the foundations of linear elastic fracture mechanics – Griffith (1921) – , using a global energy balance approach. About 90 years later, Chen developed a thermodynamics based version of Griffith's method, which: (1) considers processes in energy picture, (2) starts from global energy balance and the global form of the 2 nd law of thermodynamics, resulting in a nonlinear field theory of fracture mechanics allowing to address strongly coupled multi-field problems, including bulk dissipation – see Chen and Mai (2013) – . Below, along the lines of the general theory, an outline of a theoretical model is illustrated that seems promising for solving coupled thermo-visco-elastoplastic problems. The underlying theoretical framework used in development of the model is a form of the Thermomechanics with Internal Variables ( TIV ). Given a solid body, its deformation in the ambient space is described by the mapping ( ) ( ) , t t = x χ X , x denoting the positions in Euler and X in Lagrange coordinates,  denotes the gradient in Euler, X  in Lagrange picture. 1 2 = − E C I is the Green-Lagrangian strain tensor. t d = A A is Lagrange time derivative of a field, t d = v u means velocity. 1 t t T j − − = Σ F σ F is the second Piola-Kirchhoff stress tensor. Following the line of thought of Chen and Mai (2013), the governing equations – i.e., the balances and the constitutive laws – of the model are as follows: = − u x X is the displacement vector, =  X F χ is deformation gradient, j =det( F ), T = C F F is right Cauchy-Green tensor, ( )

t    t

0

balanceof mass

d dV 

=

t

V

V 



balanceof linear momentum

d dV  v

σ n

f

dA

dV

= 

+



t

ext

V

V



t

t

(

)

(

)

(

)

V 



balanceof angular momentum

dV  =   r v r σ n 

 r f

d

dA

dV

+ 

t

ext

V

V



t

t

t

st energy balance equation (1 law of thermodynamics)

(

)

(

)

(

)

(

)









: σ v

dV = −  +  +  j σ f 

v

d E E  +

E dV

dV

dV

+

+



t

kin

H th

X q

X

ext

X

V

V

V

V

t

t

t

t

nd dissipation requirement (2 law of thermodynamics)

(

)

(

)

(

)

(

)



V 





(4)

j

T dA j n

j

T d s dV 

T dV sd T dV 

= − − 

−  −

+

t i

q

s

s

X

t

V

V

V



t

t

t

t

(

)







1

0

: Σ C t

v v

−

dV j +

d dV d E E d  − + V

 + 



1 2

t

t

kin

H

V

V

V

t

t

t

with:

(

)

; , , E K A E  = =  kin j

the kinetic energy density

i v α X C

(

)

, , T T A X i

, , ,

the Helmholtz free energydensity the thermal energy density

α X

H

j

E T s = 

th

and the constitutive relations: 2 , , t s    =  =  Σ

(

)



,

g

G

K dV +

0  = − 



= −

 

0

C

α

T

i

j

A

i

j

V

t