PSI - Issue 42

Tamás Fekete et al. / Procedia Structural Integrity 42 (2022) 1467–1474 Tamás Fekete, Éva Feketéné-Szakos / Structural Integrity Procedia 00 (2019) 000–000

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4

RVE conjugated to P is shown below. For this, the motion of a point P' –belonging to the RVE , with coordinates d ′ = + X X X – is investigated. In other words, the behavior of the equation of motion during the transformation , d ′ ′ → → = + P P X X X X is examined. It is assumed that on the RVE , the motion field can be described by the second-order Taylor series expansion over P with sufficient accuracy. Thus, the motion of P' in ECS is as follows: ( ) ( ) ( ) ( ) ( ) 1 2 ˆ ˆ ˆ ˆ , , , , , , , : X X X d d d O d d d τ τ χ τ τ χ τ τ χ τ τ ′ = +∇ + ∇ ∇ ⊗ + ⊗ ⊗ x X X X X X X X X X (2) ) O d d d ⊗ ⊗ X X X third and higher order terms are concealed. Keeping only the locally linear part of the material deformation gradient, the classical Boltzmann continuum is obtained; in the following, the description follows the Boltzmann continuum model. The displacement vector connecting the positions of P between the ECS and the MCS is defined by: ( ) ( ) ( ) ˆ ˆ ˆ , , , τ τ τ τ τ τ = − u x X (3) The right Cauchy-Green tensor C is given by T = C F F . Therefore, the Green-Lagrange strain tensor is defined as ( ) 1 2 = − C I ε –see Béda, Kozák, Verhás (1995)–. Considering the displacement vector u as a primary variable, the displacement gradient X X = ∇ = ∇ H u u  ɺ appears in the picture, and is connected to F through = − H F I . Then, the Green-Lagrange deformation tensor takes the following form –see e.g., Béda, Kozák, Verhás (1995)–: ( ) ( ) ( ) 1 2 ˆ ˆ , , T T τ τ τ τ = + + ⋅ H H H H ε (4) The system behavior is governed by the equations of motion coupled with the field equations. The field equations include the general balances that apply to all material models and the constitutive equations governing the behavior of a particular material –see e.g., Béda, Kozák, Verhás (1995)–. For NLFTFM –see Chen, Mai (2013)–, the field equations are split into two groups: the first group describes volumetric phenomena, and the second group describes the behavior of cracks, as follows: where x ’ denotes the position of P ' in ECS , bilinear part of the material deformation gradient at X ; in ( ˆ ( , , ) X χ τ τ ∇ X is the locally linear, and ˆ ( , , ) X X χ τ τ ∇ ∇ X is the locally

( ) ( ) ( ) ( )

t    t

ρ

=

0

a themass balance

d dV

t

V

f ⌢





σ n

ρ

= ⋅

+

ρ

b the linear momentum balance

d dV v

dA

dV

t

ext

∂

V

V

V

t

t

⌢

(

)

(

)

(

)





dV × = × ⋅ r v r σ n ρ

+ ×

ρ

c the angular momentum balance

r f

d

dA

dV

t

ext

∂

V

V

V

t

t

t

st

(1 law of Thermodynamics)

d the energy balance

⌢

⌢ ⌢

⌢ (

(

)

)

(

)

(

)









X q dV = − ∇ + ∇ + ⋅ j σ f ρ X ext

: σ v

ρ

+ +

+

∇

v

d E

H th E E dV

dV

dV

X

t

kin

V

V

V

V

t

t

t

t

( )

(5)

nd

(2 law of Thermodynamics)

e thedissipation inequality

⌢

⌢

(

)

(

)

(

)

(

)









ρ

= − − ⋅ j

− ⋅∇ − j

ρ

+

T dA j n

T d s dV

T dV sd T dV

t i

q

s

s

X

t

∂

V

V

V

V

t

t

t

t ⌢ (

⌢

)

ε







Σ

−

+

ρ

⋅

+

−

ρ

+

≥

1

0

:

1 2

t

d dV j v v

d dV d E

E dV

t

t

t

kin

H

V

V

V

t

t

t

with:

⌢

⌢

(

)

( ) ( ) ( )

; , , E K A v α ɶ ɶ =

f the kinetic energy density

X

kin

i

j

E ⌢

Ψ ⌢

(

)

ε

, , T T A ∇ α ɶ ɶ X i

=

, , ,

g theHelmholtz free energydensity h the thermal energy density

X

H

j

⌢

⌢

E T s = ⋅

th