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

782 4

(

) ( ) , q x 

( ) , 

( ) ,   +  = T x

Fourier equation to heat conduction

c T x  

v



(

)

( ) ( ) , ,  

( ) , 

( ) , 

(1)

boundaryconditions at the fluid-solid interface

T x

h x T x

T x





= −

−

n

bou

bou

tflu

( ) = ,

( ) T c c T =

( ) T

,

equations of state

 

  =

v

v

The continuum mechanics part describes the motion, and the stress-strain state of the system wall as follows: ( ) ( ) ( ) , , x u u x    =  +

kinematicequation strain decomposition theorem

1 2

( ) ,         = + + +  E d d d d v x x T p

( ) ( ) , , x

( ) , 

balance of momentum

f x

=

(2)

( ) , x     = T

balance of moment of momentum balance of mechanical power

( ) ,

( ) ( ) , : , x x     

W x

= = (

(

)

) , , ) , , p p T d F T d F T     = = , T T (

constitutive relations

d F 



E

E

The fracture mechanics part consists of determining the crack tip driving forces and assessing its stability. Nowadays, classic fracture mechanics methods – see e.g., Cherepanov (1967) and Rice (1968) – are mainly used here. ( ) - determination of the crack-tip driving force stability criterion: when 0 the crack-tip is considered stable when 0 the crack-tip is considered instable T I E I Ic I Ic J W NdA J J J J    =  −  −   I (3) In the equations, x denotes space coordinate, τ indicates time, ∂ τ or ˙ represents time derivation,  is spatial gradient operator, T means temperature, q indicates power of the volumetric heat source density, ρ is mass density, c v denotes heat capacity, λ is heat conduction coefficient, ∂ n means local outer normal gradient operator at a point of an interface, h symbolizes heat-transfer coefficient, T │ bou denotes temperature of the solid and T │ tflu symbolizes effective temperature of the technological fluid at a point at the heat-transfer interface, d means differential increment of a given quantity, u denotes displacement, ε is the strain tensor, σ represents the stress tensor, ̇ v means acceleration, f denotes densities of external forces, W is the amount of stored mechanical energy in the material, W denotes mechanical power, ∙ is inner multiplication of two vectors, : means inner multiplication of two tensors. d E ε = F ( σ E , T ) represents the elastic, d T ε = F ( σ T , T ) the thermal-mechanic, d p ε = F ( σ p , T ) the plastic constitutive laws. J I denotes the crack-tip driving force and J Ic represents fracture toughness. During the design of an LSPS , in DSCs , the material laws and the ageing behaviour of structural materials are defined by the regulations; the stability of the system is evaluated under the stability criteria specified in the safety standard. In SICs for an in-service LSPS , the material laws, in conjunction with the ageing behaviour of structural materials are derived from experimental results of the equipment surveillance programme, thus taking into account their unique properties. The stability of the system is assessed in the same way, according to the stability criteria of the safety standard, as for DSCs . A deep evaluation of the methodology is beyond scope of this paper. However, its essence can be summarised as follows: (1) the theory of heat conduction and continuum mechanics have been developed largely independently, which have not been properly harmonised to date; (2) the fracture mechanics used in these calculations is based on a fairly simple continuum mechanics model; (3) the thermodynamic consistency of the whole model may not always be guaranteed; (4) for many issues concerning materials behaviour, the solutions – e.g. the description of ageing – depend almost exclusively on experimental data.