Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Thermo-cohesive crack model Jan Jaśkowiec
15 January 2015
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Introduction The considered domain has inner discontinuity Γd The domain body is subjected to mechanical and thermal load The displacement field is discontinuous along Γd and so the temperature field Along Γd the cohesion tractions are modeled Along Γd the heat transport through the crack is modeled The crack heat transport is modeled by conductivity and radiation In the crack heat transport the current crack opening vector regarded (the normal and sliding part) The dependence on temperature of the cohesion tractions and material parameters is regarded
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Domain with inner discontinuity Γ nd
Γd sd
Ω
Inside the body the crack exists Γd . On Γd the displacement vector u and temperature Θ are discontinuous.
[[·]] – the jump operator, ⟨·⟩ – the mid-value operator [[f ]] (x) = lim f (x + λnd ) − lim f (x − λnd ) = f + (x) − f − (x) λ→0
λ→0
(1)
) 1( + f (x) + f − (x) ⟨f ⟩ (x) = 2
[[f g ]] = [[f ]]⟨g ⟩ + ⟨f ⟩[[g ]] J. Jaśkowiec, L-5
Thermo-cohesive crack model
(2)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Momentum balance
The equilibrium equation (momentum balance) is valid at each point of the considered isotropic solid Ω and for each moment of time t, completed with boundary conditions: div σ + b = 0 in Ω ˆ ˆ σ· n = t on Γσ , u = u
J. Jaśkowiec, L-5
on Γu
Thermo-cohesive crack model
(3)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Weak form of the momentum balance ∫
∫ vu · div σ dΩ +
Ω
vu · b dΩ = 0
∫
∫
Ω
∇s vu : σ dΩ + Ω
∫
(5)
[[σ· vu ]]· nd dΓ
(6)
∫ (σ· vu ) · n dΓ −
div (σ· vu ) dΩ =
vu · b dΩ = 0 Ω
∫ Γ
Γd
[[σ]] = 0 , ∫
σ· nd = td
on Γd
∫
∫
div (σ· vu ) dΩ = Ω
(4)
∫
div (σ· vu ) dΩ −
Ω
∀ vu
Ω
J. Jaśkowiec,
Γ L-5
vu · t dΓ −
[[vu ]]· td dΓ
Γd Thermo-cohesive
crack model
(7)
(8)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Weak form of the momentum balance
∫
∫
Ω
∫ [[vu ]]· td dΓ −
ε(vu ) : σ dΩ + Γd
∫ vu · b dΩ −
Ω
vu · t dΓ = 0 Γ
ε(vu ) = ∇s vu td – vector of cohesion tractions along crack surface
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(9)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Energy balance
The thermal model starts with the well-known form of energy balance ˙ + div q = r , cρΘ in Ω ˆ on ΓΘ , q· n = hˆ on Γh , Θ=Θ
Θ(t0 ) = 0 in Ω
ρ – material density, c – specific heat capacity, Θ – temperature value in relation to reference temperature T0 , q – heat flux density, r – heat source density.
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(10)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Weak form of energy balance ∫
∫
Ω
∫ vΘ div q dΩ −
˙ dΩ + vΘ cρΘ Ω
∫
∫
Ω
∫ div (vΘ q) dΩ −
˙ dΩ + vΘ cρΘ Ω
(11)
∫ ∇vΘ · q dΩ −
Ω
∫
∫ vΘ q· n dΓ −
div (vΘ q) dΩ = Ω
Γ
[[vΘ q]]· nd dΓ
(13)
Γd
[[q]] = 0 , ∫
q· nd = hd
on Γd
∫
∫
div (vΘ q) dΩ = J. Jaśkowiec,
vΘ r dΩ = 0 (12) Ω
∫
Ω
∀vΘ
vΘ r dΩ = 0 , Ω
Γ L-5
vΘ h dΓ −
[[vΘ ]]hd dΓ
Γd Thermo-cohesive
crack model
(14)
(15)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Weak form of energy balance
∫
∫
Ω
∫
−
∫ vΘ h dΓ −
˙ dΩ + vΘ cρΘ Γ
∫
∇vΘ · q dΩ − Ω
[[vΘ ]]hd dΓ Γd
vΘ r dΩ = 0 Ω
hd –the crack heat flux (heat stream that goes through the crack surface)
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(16)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
2D crack subspace The crack opening vector [[u]] can be decomposed into a normal part and a sliding part, where the normal part of the vector is obtained with: wn = nd · [[u]] {
wn > 0 the crack opens outward wn < 0 the crack penetrates the body
(17)
(18)
Then the normal crack opening vector is given as: wn = nd wn = nd ⊗ nd · [[u]]
(19)
which means that the vector wn is the projection of [[u]] in the direction of nd . J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
2D crack subspace The so-called sliding crack opening vector is calculated as the difference of [[u]] and wn : ws = [[u]] − wn ws = [[u]] − nd ⊗ nd · [[u]]
(20) (21)
ws = (I − nd ⊗ nd ) · [[u]]
(22)
This means that ws is the projection of [[u]] onto the crack surface Γd . Subsequently, the unit vector in the sliding direction is given by sd =
ws ∥ws ∥
(23)
The magnitude of the sliding part of crack opening vector then becomes ws = sd · [[u]] J. Jaśkowiec, L-5
Thermo-cohesive crack model
(24)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
2D crack subspace
The vectors nd and sd define a 2D local coordinate frame. The normal vector to the crack nd is connected with the crack geometry and therefore constant in time (although it may vary in space). The sliding vector sd depends on the crack opening vector [[u]] and may therefore change in time.
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Nonlinear components of the mathematical model In the mathematical model the nonlinear components are cohesion tractions td and crack heat flux hd since they depend on the current displacement and temperature states. It is assumed that the dependence can be expressed in the forms td = td ([[u]], Θd )
(25)
hd = hd ([[u]], [[Θ]], ⟨∇Θ⟩)
(26)
The Θd is the temperature in the crack surface. In the paper the temperature Θd is calculated as a mid temperature between both sides of the crack Θd = ⟨Θ⟩ + T0 J. Jaśkowiec, L-5
Thermo-cohesive crack model
(27)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Linearization of cohesion vector and crack heat flux
The linearisation gives the following relations: ( ) ∂td ∂td δtd = · δ[[u]] + δΘd ∂[[u]] ∂Θd ( δhd =
∂hd ∂[[u]]
) · δ[[u]] +
∂hd ∂hd δ[[Θ]] + · δ⟨∇Θ⟩ ∂[[Θ]] ∂⟨∇Θ⟩
where here δ means small (corrective) increment parameter.
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(28)
(29)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Linearization of cohesion vector The crack opening vector and the cohesion tractions vector can be decomposed into normal to the crack and crack sliding direction sd ,
The derivatives
∂td ∂[[u]]
and
[[u]] = nd wn + sd ws
(30)
td = nd tn + sd ts
(31)
∂td ∂Θd
can be effectively written in (nd , sd )
∂td ∂tn ∂ts ∂sd = nd + sd + ts ∂[[u]] ∂[[u]] ∂[[u]] ∂[[u]]
(32)
∂td ∂tn ∂ts ∂sd = nd + sd + ts ∂Θd ∂Θd ∂Θd ∂Θd
(33)
∂sd =0 ∂Θd
(34)
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Linearization of cohesion vector
∂tn ∂tn ∂wn ∂tn ∂ws ∂tn ∂tn = + = nd T + sd T ∂[[u]] ∂wn ∂[[u]] ∂ws ∂[[u]] ∂wn ∂ws ∂ts ∂ts ∂wn ∂ts ∂ws ∂t ∂ts s = + = nd T + sd T ∂[[u]] ∂wn ∂[[u]] ∂ws ∂[[u]] ∂wn ∂ws ∂sd ∂(ws /ws ) ws (I − nd nd T ) − ws sd T = = ∂[[u]] ∂[[u]] ws 2 T T I − nd nd − sd sd 1 = = · rd ⊗ rd ws ws where rd = nd × sd .
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(35) (36)
(37)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Linearization of crack heat flux The derivative of hd in respect to crack opening vector [[u]] takes the form ∂hd ∂hd ∂wn ∂hd ∂ws ∂hd ∂sd = + + ∂[[u]] ∂wn ∂[[u]] ∂ws ∂[[u]] ∂sd ∂[[u]]
(38)
∂hd ∂hd 1 ∂hd = Q· + · (rd ⊗ rd ) · ∂[[u]] ∂wns ws ∂sd
(39)
where [ Q = nd
sd
[
wns J. Jaśkowiec, L-5
w = n ws
]
(40)
]
Thermo-cohesive crack model
(41)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Time integration and physical relations The backward Euler time integration is applied ˙ t+∆t → Θ ˙ t+∆t = ∆Θ Θt+∆t = Θt + ∆t Θ ∆t
(42)
The total strain tensor can be decomposed into elastic εe and thermal εΘ parts: ε = εe + εΘ
where εΘ = αΘI
(43)
The Hook’s stress–strain relation is applied δσ = E : δεe = E : δε − e δΘ where e = −αE : I
(44)
The Fourier law: δq = −Λ∇( δΘ) J. Jaśkowiec, L-5
Thermo-cohesive crack model
(45)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Linearized weak forms ∫
∫ ε(vu ) : E : δε(u) dΩ +
V
∫
∂td [[vu ]]T δ[[u]] dΓ + ∂[[u]]
Γd
∫
ε(vu ) : e δΘ dΩ+ V
[[vu ]]T
∂td δΘd dΓ + = Wt+∆t − Wt+∆t ext int,i ∂Θd
Γd
(46) and
∫ ∫ 1 T vΘ cρ δΘ dΩ + (∇vΘ ) Λ∇( δΘ) dΩ− ∆t Ω Ω ∫ ∫ ∂hd ∂hd [[vΘ ]] · δ[[u]] dΩ − [[vΘ ]] δ[[Θ]] dΩ− ∂[[u]] ∂[[Θ]] Γd Γd ∫ ∂hd t+∆t t+∆t · δ⟨∇Θ⟩ dΩ = Qext − Qint,i [[vΘ ]] ∂⟨∇Θ⟩ Γd J. Jaśkowiec, L-5
Thermo-cohesive crack model
(47)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Modeling of heat flow through crack
In the model the following combinations of heat transfers through the crack is taken into account: heat transfer by conduction of the gas filling the crack, heat transfer by conduction of the bridging elements of the material, heat transfer by radiation, It is supposed that the crack heat flux is superposition of heat flux by conduction hdc and heat flux by radiation hdr , and so their increments δhd = δhdc + δhdr
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(48)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat conduction through crack In the model the crack opening and crack sliding are applied
hdc = −κ
[[Θ]]s wn
(49)
where κ is the conductivity parameter of the medium between crack faces ( ) 1 1 [[Θ]]s =Θ+ + · ∇Θ+ · ws − Θ− − · ∇Θ− · ws = 2 2 (50) [[Θ]] + ⟨∇Θ⟩· ws The sliding part of crack opening vector ws can be expressed by crack multiplication of sliding direction sd and the value of crack sliding ws ⟨ ⟩ [[Θ]]s = [[Θ]] + ∇Θ · sd ws (51) J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat conduction through crack The derivatives of hdc can now be evaluated ∂hdc κ =− ∂[[Θ]] wn ws ∂hdc = −κ sd ∂⟨∇Θ⟩ wn ∂hdc [[Θ]]s =κ 2 ∂wn wn c ∂hd ⟨∇Θ⟩· sd = −κ ∂ws wn ∂hdc κws =− ⟨∇Θ⟩ ∂sd wn
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(52a) (52b) (52c) (52d) (52e)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat conduction through crack When taking into account evaluated derivatives in (52) then the derivative of hdc in relation to crack opening vector [[u]] reads as: ∂hdc [[Θ]]s κ κ = nd κ 2 − (sd ⊗ sd ) · ⟨∇Θ⟩ − (rd ⊗ rd ) · ⟨∇Θ⟩ ∂[[u]] wn wn wn
(53)
In the 3D case the following relation is satisfied: nd ⊗ nd + sd ⊗ sd + rd ⊗ rd = I
(54)
When taking into account relation from (54) to (53) it goes to the following form ) ∂hdc [[Θ]]s κ ( = nd κ 2 + · nd ⊗ nd − I · ⟨∇Θ⟩ ∂[[u]] wn wn
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(55)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat conduction through crack Remembering that δ⟨∇Θ⟩ = ⟨∇ δΘ⟩, then the corrective increment of hdc from equation (29) can now be expressed as follows ( ) ) [[Θ]]s ( κ κ c δhd = nd + nd ⊗ nd − I ⟨∇Θ⟩ · δ[[u]] − δ[[Θ]]s wn wn wn ws =const
(56) what finally can be expressed as: c c ∂h ∂h d δhdc = d · δ[[u]] + δ[[Θ]]s ∂[[u]] ∂[[Θ]]s
ws =const
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(57)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat conduction through crack
The κ parameter has the interpretation of the crack conductivity . It is the combination of air conductivity and bridging elements of the material. It is assumed that when the crack opens, at the beginning of the process some element of the materials bridges the crack faces. It can be written in the following way κ = (1 − d)κm + dκa where κm is the thermal conductivity of the material, κa thermal conductivity of the air, d is a crack damage parameter.
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(58)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat radiation through crack The model of the radiation crack heat flux hdr is based on Stefan-Boltzmann law. It is supposed in the paper that the amount of heat radiated out by hotter crack face is absorbed by the other crack face. In that case the radiation heat flux can be written as ( ) hdr = eσ · (Ts+ )4 − (Ts− )4 (59) Ts+ and Ts− – are absolute temperatures at both sides of crack regarding crack sliding, e – is the emissivity parameter 0 < e ≤ 1, σ – is Stefan’s constant (σ = 5.678 · 10−8 [W/m2 K]). The radiation heat flux can be expressed as follows: ( ) ( ) ( ) (60) hdr = eσ · (Ts+ )2 + (Ts− )2 · Ts+ + Ts− · Ts+ − Ts− and
( ) ( ) hdr = eσ · (Ts+ )2 + (Ts− )2 · Ts+ + Ts− · [[Θ]]s J. Jaśkowiec, L-5
Thermo-cohesive crack model
(61)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat radiation through crack
The following function of absolute temperatures on both sides of crack faces can be defined here ( ) ( ) H r = −eσ · (Ts+ )2 + (Ts− )2 · Ts+ + Ts− (62) then the heat flow through radiation is expressed by the relation similar to convecitve interface law hdr = −H r [[Θ]]s
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(63)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat radiation through crack The change of hdr is connected with changes of temperatures differences on both sides of the crack. The change of temperature jump is due to change of [[Θ]] and change of crack sliding ws . It can be written as ∂hdr ∂hdr r δhd = δ[[u]] + δ[[Θ]]s (64) ∂[[u]] ∂[[Θ]]s ws =const
( ∂[[Θ]] ∂hdr ∂hdr ∂[[Θ]]s 1 ∂[[Θ]]s ) s = = −H r · sd + · (rd ⊗ rd ) ∂[[u]] ∂[[Θ]]s ∂[[u]] ∂ws ws ∂sd (65)
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat radiation through crack Taking into account the definition of [[u]]s in (51) it can be easily written that ∂[[Θ]]s = ⟨∇Θ⟩· sd , ∂ws
∂[[Θ]]s = ⟨∇Θ⟩ws ∂sd
(66)
After incorporating (66) to (65) it result in ( ) ∂hdr = −H r · sd ⊗ sd + rd ⊗ rd ⟨∇Θ⟩ ∂[[u]]
(67)
Finally, after using the relation (54), the increment of radiation crack heat flux has the form ( ) r r r (68) δhd = −H · I − nd ⊗ nd ⟨∇Θ⟩ δ[[u]] − H δ[[Θ]]s ws =const
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat flow through crack ( hd = −
κ + Hr wn
) (69)
[[Θ]]s
The change of crack heat flux is due to change of crack opening and change of temperature jump between both crack faces ∂hd ∂hd δhd = δ[[u]] + δ[[Θ]]s (70) ∂[[u]] ∂[[Θ]]s ws =const
The derivatives in (70) can now be written as ( ) ∂hd κ =− + Hr ∂[[Θ]]s wn ) ( ( ) [[Θ]]s ∂hd κ = nd κ 2 + + H r · nd ⊗ nd − I ⟨∇Θ⟩ ∂[[u]] wn wn J. Jaśkowiec, L-5
Thermo-cohesive crack model
(71) (72)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
The modified Xu-Needleman cohesion law is used ( ) ( ) GI ws 2 wn wn tn = F · exp − exp − wnc wnc wnc wsc 2 ( ) ( ) GIIF ws ( wn ) ws 2 wn ts = 2 · · 1− exp − exp − wsc wsc wnc wsc 2 wnc
(73)
where GIF tnc exp(1) GII = √ F c ts 12 exp(1)
wnc = wsc
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(74)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
It is assumed that the cohesion law parameters: tnc , tsc , GIF , GIIF depend on temperature ( ) tn = tn wn , ws , tnc (Θd ), GIF (Θd ), tsc (Θd ), GIIF (Θd ) ( ) (75) ts = ts wn , ws , tsc (Θd ), GIIF (Θd ), tnc (Θd ), GIF (Θd ) In result the derivatives from (33) can now be written using the chain rule ∂tn ∂tn ∂t c ∂tn ∂GIF ∂tn ∂t c ∂tn ∂GIIF = c n + + c s + ∂Θd ∂tn ∂Θd ∂GIF ∂Θd ∂ts ∂Θd ∂GIIF ∂Θd ∂ts ∂tsc ∂ts ∂GIIF ∂ts ∂tnc ∂ts ∂GIF ∂ts = c + + c + ∂Θd ∂ts ∂Θd ∂GIIF ∂Θd ∂tn ∂Θd ∂GIF ∂Θd
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(76)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
∂tn ∂wnc ∂tn = ∂tnc ∂wnc ∂tnc ( w ) ( w2) GI wn − 2wnc n s = − cF wn · exp − exp − 3 tn wnc wnc wsc 2 ∂tn ∂tn ∂tn ∂wnc = + ∂GIF ∂GIF ∂wnc ∂GIF wnc =const
( w ) ( w2) wn − wnc n s = wn exp − exp − 3 wnc wnc wsc 2 ∂tn ∂tn ∂wsc = ∂GIIF ∂wsc ∂GIIF ( ) ) ( wn wn 2ws 2 GIF ws 2 · exp − = · exp − wsc 2 GIIF wnc wnc wnc wsc 2 ∂tn ∂tn ∂wsc = ∂tsc ∂wsc ∂tsc ( ) ( ) 2ws 2 wn wn GIF ws 2 =− · · exp − exp − wsc 2 tsc wnc wnc wnc wsc 2 J. Jaśkowiec, L-5
Thermo-cohesive crack model
(77)
(78)
(79)
(80)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
∂ts ∂wsc ∂ts = ∂tsc ∂wsc ∂tsc ( ) ( ) (81) GII wsc 2 − ws ( wn ) ws 2 wn = 4 cF ws · 1 − exp − exp − ts wsc 4 wnc wsc 2 wnc ∂ts ∂ts ∂ts ∂wsc = + ∂GIIF ∂GIIF ∂wsc ∂GIIF wsc =const ( ) ( ) wn ) ws 2 wn ws wsc − 2ws 2 ( · 1 − exp − exp − = −2 wsc 3 wnc wsc 2 wnc (82) ∂ts ∂ts ∂wnc = ∂tnc ∂wnc ∂tnc ( ) ( ) GIIF wnc ws 2wn wnc − wn 2 ws 2 wn = −2 c · · exp − exp − tn wsc wsc wnc 3 wsc 2 wnc (83) ∂ts ∂ts ∂wnc = ∂GIF ∂wnc ∂GIF J. Jaśkowiec, L-5 Thermo-cohesive ( crack model ) ( )
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
XFEM interpolation In order to incorporate discontinuities in the displacements and temperature fields, XFEM approximation us used. The displacement vector and temperature field are approximated in a standard way: δu = Φu δˇ u,
ˇ δΘ = ΦΘ δ Θ
(85)
where Φu and ΦΘ are the approximation matrices for displacement and temperature fields respectively. The jumps of displacement vector and temperature are approximated by their approximations jumps: δ[[u]] = [[Φu ]] δˇ u
ˇ δ[[Θ]] = [[ΦΘ ]] δ Θ
(86)
The test functions, i.e. vu , vΘ , and their jumps are approximated in the same manner. The temperature gradient temperature approximation ˇ = BΘ δ Θ ˇ ∇ δΘ = ∇ΦΘ δ Θ J. Jaśkowiec, L-5
Thermo-cohesive crack model
(87)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
XFEM interpolation The crack temperature mid-gradient can be approximated in simmilar manner ˇ ⟨∇ δΘ⟩ = ⟨BΘ ⟩ δ Θ
(88)
( ) − where ⟨BΘ ⟩ = 0.5 B+ Θ + BΘ . The temperature jump increment in situation when the crack sliding is taken into account can now be approximated in the following way: ( ) ˇ = [[ΦΘ ]]s δ Θ ˇ δ[[Θ]]s = [[ΦΘ ]] + wsT ⟨BΘ ⟩ δ Θ (89) ws =const
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Linearized system of equations [
K + Kd Hu
KΘ + KΘ d H + Hd
][
] [ t+∆t ] Rf int,i δˇ u ˇ = Rh t+∆t δΘ int,i
where t+∆t t+∆t Rf t+∆t − fint,i int,i = fext t+∆t t+∆t Rh t+∆t int,i = hext − hint,i
∫
∫ BT u EBu dΩ ,
K= Ω
Γd
∫ BT u e ΦΘ dΩ ,
KΘ =
[[Φu ]]T
Kd =
∫ [[Φu ]]T
KΘd =
Ω
∂td [[Φu ]] dΓ , ∂[[u]] ∂td ⟨Φd ⟩ dΓ , ∂Θd
Γd J. Jaśkowiec, L-5
Thermo-cohesive crack model
(90)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Linearized system of equations (
∫ Hu = −
[[ΦΘ ]]T
∂hd ∂[[u]]
)T
(
∫ Hd = −
[[Φu ]] dΩ ,
Ω
[[ΦΘ ]]T
∂hd ∂[[Θ]]
) [[ΦΘ ]] dΓ ,
Γd
1 H= ∆t
∫
∫ ΦT Θ cρΦΘ
Ω
Ω
∫
∫ t+∆t fint,i
t+∆t BT u σi
=
Γd
∫
∫ ΦT Θ cρ
t+∆t BT dΩ + Θ qi Ω
dΓ , [[Φu ]]T td t+∆t i
dΩ +
Ω
t+∆t hint,i =−
BT Θ ΛBΘ dΩ ,
dΩ +
Ω J. Jaśkowiec, L-5
∆Θi dΩ − ∆t
∫ [[ΦΘ ]]hd t+∆t dΓ i Γd
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
The mesh updating Let us suppose that there are two interpolation matrices connected with two different meshes: Φ1Θ and Φ2Θ . The temperature field can be approximated by either of the approximation matrices: ˇ 1 ≈ Φ2Θ Θ ˇ2 Θ ≈ Φ1Θ Θ
(91)
ˇ 1 is known vector of degrees of freedom for the first mesh and where Θ ˇ Θ2 is the vector of degrees of freedom for the updated mesh that have to be calculated. An error between those two approximations is defined in the following way ∫ ( 1 ) ˇ 1 − Φ2Θ Θ ˇ 2 2 dΩ ΦΘ Θ ϵΘ = (92) Ω
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
The mesh updating ˇ 2 have to be chosen so that to minimizes the error that The vector Θ leads to the equation: ∫ ) ∂ϵΘ T( ˇ 1 − Φ2Θ Θ ˇ 2 dΩ = 0 =0 ⇒ Φ2Θ Φ1Θ Θ (93) ˇ ∂ Θ2 Ω
Eventually the vector of the degrees of freedom for the updated mesh is obtained by solving the equation ˇ 2 = M21 Θ ˇ1 M2 Θ
(94)
where M2 is the mass matrix for the updated mesh and M21 is the mass matrix based on the two meshes: the previous mesh and the current mesh. J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
Material parameters Property Density Poisson’s ratio Modulus of elasticity Coeff. of ther. expansion Specific heat capacity Thermal conductivity Reference temperature Tensile strength Fracture energy for mode I Shear strength Fracture energy for mode II emissivity parameter
Symbol ρ ν E α c λΘ T0 tnc GIF tsc GIIF e
Value 2300 0.2 27 14.5 · 10−6 0.75 1.8 20 3 0.08 3 0.08 2 · 10−3
Unit kg/m3 GPa 1/K kJ/(kg K) W/(m K) [o C] MPa kJ/m2 MPa kJ/m2
Table: Material parameters for the examples. The values are given for the reference temperature
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
Example 1 F
F
qn qn = 5 [kW/mm2 ] ˆ = 0 deg Θ
10cm ˆ Θ
3cm 8cm Figure: The geometry and boundary conditions for example 1 J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
Example 1
Figure: Temperature distribution in deformed structure without crack sliding modeling
J. Jaśkowiec, L-5
Figure: Temperature distribution with crack sliding modeling
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
Example 2 F F 10cm
qn = 5 [kW/mm2 ]
qn ˆ Θ
ˆ = 0 deg Θ
3cm 8cm Figure: The geometry and boundary conditions for example 2 J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
(a) Temperature without radiation
(b) Heat flux without radiation
(c) Temperature with radiation
(d) Heat flux with radiation
Figure: Temperature and heat flux maps for the example 2 with 0.1mm CMOV J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
(a) Temperature without radiation
(b) Heat flux without radiation
(c) Temperature with radiation
(d) Heat flux with radiation
Figure: Temperature and heat flux maps for the example 2 with 0.5mm CMOV J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
(a) Temperature without radiation
(b) Heat flux without radiation
(c) Temperature with radiation
(d) Heat flux with radiation
Figure: Temperature and heat flux maps for the example 2 with 2mm CMOV J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
(a) Temperature without radiation
(b) Heat flux without radiation
(c) Temperature with radiation
(d) Heat flux with radiation
Figure: Temperature and heat flux maps for the example 3 with 2mm and 8mm CMOV J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
The crack growth example ( tnc (Θd ) = tnc T0 1 −
Θd ) 580[o C] Θd GIF (Θd ) = GIF T0 (1 + ) 400[o C] ( 3 Θd ) · tsc (Θd ) = tsc T0 1 − 10 400[o C] Θd GIIF (Θd ) = GIIF T0 (1 + ) 400[o C]
(95)
In the continuum the Young modulus E and thermal conductivity λΘ parameters also change with temperature in the following way Θ E (Θ) = ET0 (1 − ) 600[o C] (96) Θ 1 λΘ (Θ) = λΘ T0 (1 − · ) 2 800[o C] J. Jaśkowiec, L-5 Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
Example 2 F F 10cm
qn = 5 [kW/mm2 ]
qn ˆ Θ
ˆ = 0 deg Θ
3cm 8cm Figure: The geometry and boundary conditions for example 4 J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
(a) Thermo-mechanical model
Material parameters Example 1 Example 2 Example 3 Example 4
(b) Mechanical model
Figure: The configuration in the thermo-mechanical analysis after 2.3 s
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
(a) Thermo-mechanical model
Material parameters Example 1 Example 2 Example 3 Example 4
(b) Mechanical model
Figure: The configuration in the thermo-mechanical analysis after 5 s
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
(a) Thermo-mechanical model
Material parameters Example 1 Example 2 Example 3 Example 4
(b) Mechanical model
Figure: The configuration in the thermo-mechanical analysis after 13 s
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Conclusions The existence of the crack in the body has an influence on heat flow and temperature distribution The 3D model of heat flow through the crack has been presented that takes into account crack opening, crack sliding, radiation and bridging The model is described in local 2D subspace (nd , sd ), i.e. crack normal and crack sliding directions On feedback the temperature distribution has an influence on cohesion forces, crack path and crack growth criteria When the crack growths the mesh configuration is changing and the number of degrees of freedom is changing. In the incremental–iterative procedure the mesh updating is necessary The further effort is needed for full 3D example. J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References J. Jaśkowiec. Three-dimensional analysis of a cohesive crack coupled with heat flux through the crack. Advances in Engineering Software, page 0, 2015. - accepted for publication. J. Jaśkowiec. Modelling of heat flow through a three-dimensional crack in thermoelasticity. In B.H.V. Topping and P. Iványi, editors, Proceedings of the Fourteenth International Conference on Civil, Structural and Environmental Engineering Computing, Stirlingshire, UK, September paper 65, 2014. doi:10.4203/ccp.105.65. Civil-Comp Press. J. Jaśkowiec. A coupled thermo-mechanical cohesive crack model in three-dimensional crack growth analysis. In B.H.V. Topping and P. Iványi, editors, Proceedings of the Fourteenth International Conference on Civil, Structural and Environmental Engineering Computing, Stirlingshire, UK, September paper 67, 2014. doi:10.4203/ccp.105.67. Civil-Comp Press. J. Jaśkowiec. Three-dimensional analysis of cohesive crack growth coupled with nonlinear thermoelasticity’. In B.H.V. Topping and P. Iványi, editors, Proceedings of the Fourteenth International Conference on Civil, Structural and Environmental Engineering Computing, Stirlingshire, UK, September paper 95, 2013. doi:10.4203/ccp.102.95. Civil-Comp Press. J. Jaśkowiec and F. P. van der Meer. A consistent iterative scheme for 2D and 3D cohesive crack analysis in XFEM. Computers & Structures, 136:98–107, 2014.
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Thermo-cohesive crack model Jan Jaśkowiec
15 January 2015
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Introduction The considered domain has inner discontinuity Γd The domain body is subjected to mechanical and thermal load The displacement field is discontinuous along Γd and so the temperature field Along Γd the cohesion tractions are modeled Along Γd the heat transport through the crack is modeled The crack heat transport is modeled by conductivity and radiation In the crack heat transport the current crack opening vector regarded (the normal and sliding part) The dependence on temperature of the cohesion tractions and material parameters is regarded
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Domain with inner discontinuity Γ nd
Γd sd
Ω
Inside the body the crack exists Γd . On Γd the displacement vector u and temperature Θ are discontinuous.
[[·]] – the jump operator, ⟨·⟩ – the mid-value operator [[f ]] (x) = lim f (x + λnd ) − lim f (x − λnd ) = f + (x) − f − (x) λ→0
λ→0
(1)
) 1( + f (x) + f − (x) ⟨f ⟩ (x) = 2
[[f g ]] = [[f ]]⟨g ⟩ + ⟨f ⟩[[g ]] J. Jaśkowiec, L-5
Thermo-cohesive crack model
(2)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Momentum balance
The equilibrium equation (momentum balance) is valid at each point of the considered isotropic solid Ω and for each moment of time t, completed with boundary conditions: div σ + b = 0 in Ω ˆ ˆ σ· n = t on Γσ , u = u
J. Jaśkowiec, L-5
on Γu
Thermo-cohesive crack model
(3)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Weak form of the momentum balance ∫
∫ vu · div σ dΩ +
Ω
vu · b dΩ = 0
∫
∫
Ω
∇s vu : σ dΩ + Ω
∫
(5)
[[σ· vu ]]· nd dΓ
(6)
∫ (σ· vu ) · n dΓ −
div (σ· vu ) dΩ =
vu · b dΩ = 0 Ω
∫ Γ
Γd
[[σ]] = 0 , ∫
σ· nd = td
on Γd
∫
∫
div (σ· vu ) dΩ = Ω
(4)
∫
div (σ· vu ) dΩ −
Ω
∀ vu
Ω
J. Jaśkowiec,
Γ L-5
vu · t dΓ −
[[vu ]]· td dΓ
Γd Thermo-cohesive
crack model
(7)
(8)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Weak form of the momentum balance
∫
∫
Ω
∫ [[vu ]]· td dΓ −
ε(vu ) : σ dΩ + Γd
∫ vu · b dΩ −
Ω
vu · t dΓ = 0 Γ
ε(vu ) = ∇s vu td – vector of cohesion tractions along crack surface
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(9)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Energy balance
The thermal model starts with the well-known form of energy balance ˙ + div q = r , cρΘ in Ω ˆ on ΓΘ , q· n = hˆ on Γh , Θ=Θ
Θ(t0 ) = 0 in Ω
ρ – material density, c – specific heat capacity, Θ – temperature value in relation to reference temperature T0 , q – heat flux density, r – heat source density.
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(10)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Weak form of energy balance ∫
∫
Ω
∫ vΘ div q dΩ −
˙ dΩ + vΘ cρΘ Ω
∫
∫
Ω
∫ div (vΘ q) dΩ −
˙ dΩ + vΘ cρΘ Ω
(11)
∫ ∇vΘ · q dΩ −
Ω
∫
∫ vΘ q· n dΓ −
div (vΘ q) dΩ = Ω
Γ
[[vΘ q]]· nd dΓ
(13)
Γd
[[q]] = 0 , ∫
q· nd = hd
on Γd
∫
∫
div (vΘ q) dΩ = J. Jaśkowiec,
vΘ r dΩ = 0 (12) Ω
∫
Ω
∀vΘ
vΘ r dΩ = 0 , Ω
Γ L-5
vΘ h dΓ −
[[vΘ ]]hd dΓ
Γd Thermo-cohesive
crack model
(14)
(15)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Weak form of energy balance
∫
∫
Ω
∫
−
∫ vΘ h dΓ −
˙ dΩ + vΘ cρΘ Γ
∫
∇vΘ · q dΩ − Ω
[[vΘ ]]hd dΓ Γd
vΘ r dΩ = 0 Ω
hd –the crack heat flux (heat stream that goes through the crack surface)
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(16)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
2D crack subspace The crack opening vector [[u]] can be decomposed into a normal part and a sliding part, where the normal part of the vector is obtained with: wn = nd · [[u]] {
wn > 0 the crack opens outward wn < 0 the crack penetrates the body
(17)
(18)
Then the normal crack opening vector is given as: wn = nd wn = nd ⊗ nd · [[u]]
(19)
which means that the vector wn is the projection of [[u]] in the direction of nd . J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
2D crack subspace The so-called sliding crack opening vector is calculated as the difference of [[u]] and wn : ws = [[u]] − wn ws = [[u]] − nd ⊗ nd · [[u]]
(20) (21)
ws = (I − nd ⊗ nd ) · [[u]]
(22)
This means that ws is the projection of [[u]] onto the crack surface Γd . Subsequently, the unit vector in the sliding direction is given by sd =
ws ∥ws ∥
(23)
The magnitude of the sliding part of crack opening vector then becomes ws = sd · [[u]] J. Jaśkowiec, L-5
Thermo-cohesive crack model
(24)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
2D crack subspace
The vectors nd and sd define a 2D local coordinate frame. The normal vector to the crack nd is connected with the crack geometry and therefore constant in time (although it may vary in space). The sliding vector sd depends on the crack opening vector [[u]] and may therefore change in time.
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Nonlinear components of the mathematical model In the mathematical model the nonlinear components are cohesion tractions td and crack heat flux hd since they depend on the current displacement and temperature states. It is assumed that the dependence can be expressed in the forms td = td ([[u]], Θd )
(25)
hd = hd ([[u]], [[Θ]], ⟨∇Θ⟩)
(26)
The Θd is the temperature in the crack surface. In the paper the temperature Θd is calculated as a mid temperature between both sides of the crack Θd = ⟨Θ⟩ + T0 J. Jaśkowiec, L-5
Thermo-cohesive crack model
(27)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Linearization of cohesion vector and crack heat flux
The linearisation gives the following relations: ( ) ∂td ∂td δtd = · δ[[u]] + δΘd ∂[[u]] ∂Θd ( δhd =
∂hd ∂[[u]]
) · δ[[u]] +
∂hd ∂hd δ[[Θ]] + · δ⟨∇Θ⟩ ∂[[Θ]] ∂⟨∇Θ⟩
where here δ means small (corrective) increment parameter.
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(28)
(29)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Linearization of cohesion vector The crack opening vector and the cohesion tractions vector can be decomposed into normal to the crack and crack sliding direction sd ,
The derivatives
∂td ∂[[u]]
and
[[u]] = nd wn + sd ws
(30)
td = nd tn + sd ts
(31)
∂td ∂Θd
can be effectively written in (nd , sd )
∂td ∂tn ∂ts ∂sd = nd + sd + ts ∂[[u]] ∂[[u]] ∂[[u]] ∂[[u]]
(32)
∂td ∂tn ∂ts ∂sd = nd + sd + ts ∂Θd ∂Θd ∂Θd ∂Θd
(33)
∂sd =0 ∂Θd
(34)
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Linearization of cohesion vector
∂tn ∂tn ∂wn ∂tn ∂ws ∂tn ∂tn = + = nd T + sd T ∂[[u]] ∂wn ∂[[u]] ∂ws ∂[[u]] ∂wn ∂ws ∂ts ∂ts ∂wn ∂ts ∂ws ∂t ∂ts s = + = nd T + sd T ∂[[u]] ∂wn ∂[[u]] ∂ws ∂[[u]] ∂wn ∂ws ∂sd ∂(ws /ws ) ws (I − nd nd T ) − ws sd T = = ∂[[u]] ∂[[u]] ws 2 T T I − nd nd − sd sd 1 = = · rd ⊗ rd ws ws where rd = nd × sd .
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(35) (36)
(37)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Linearization of crack heat flux The derivative of hd in respect to crack opening vector [[u]] takes the form ∂hd ∂hd ∂wn ∂hd ∂ws ∂hd ∂sd = + + ∂[[u]] ∂wn ∂[[u]] ∂ws ∂[[u]] ∂sd ∂[[u]]
(38)
∂hd ∂hd 1 ∂hd = Q· + · (rd ⊗ rd ) · ∂[[u]] ∂wns ws ∂sd
(39)
where [ Q = nd
sd
[
wns J. Jaśkowiec, L-5
w = n ws
]
(40)
]
Thermo-cohesive crack model
(41)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Time integration and physical relations The backward Euler time integration is applied ˙ t+∆t → Θ ˙ t+∆t = ∆Θ Θt+∆t = Θt + ∆t Θ ∆t
(42)
The total strain tensor can be decomposed into elastic εe and thermal εΘ parts: ε = εe + εΘ
where εΘ = αΘI
(43)
The Hook’s stress–strain relation is applied δσ = E : δεe = E : δε − e δΘ where e = −αE : I
(44)
The Fourier law: δq = −Λ∇( δΘ) J. Jaśkowiec, L-5
Thermo-cohesive crack model
(45)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Mathematical model of mechanical part Mathematical model of thermal part The crack local coordinates Linearisation
Linearized weak forms ∫
∫ ε(vu ) : E : δε(u) dΩ +
V
∫
∂td [[vu ]]T δ[[u]] dΓ + ∂[[u]]
Γd
∫
ε(vu ) : e δΘ dΩ+ V
[[vu ]]T
∂td δΘd dΓ + = Wt+∆t − Wt+∆t ext int,i ∂Θd
Γd
(46) and
∫ ∫ 1 T vΘ cρ δΘ dΩ + (∇vΘ ) Λ∇( δΘ) dΩ− ∆t Ω Ω ∫ ∫ ∂hd ∂hd [[vΘ ]] · δ[[u]] dΩ − [[vΘ ]] δ[[Θ]] dΩ− ∂[[u]] ∂[[Θ]] Γd Γd ∫ ∂hd t+∆t t+∆t · δ⟨∇Θ⟩ dΩ = Qext − Qint,i [[vΘ ]] ∂⟨∇Θ⟩ Γd J. Jaśkowiec, L-5
Thermo-cohesive crack model
(47)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Modeling of heat flow through crack
In the model the following combinations of heat transfers through the crack is taken into account: heat transfer by conduction of the gas filling the crack, heat transfer by conduction of the bridging elements of the material, heat transfer by radiation, It is supposed that the crack heat flux is superposition of heat flux by conduction hdc and heat flux by radiation hdr , and so their increments δhd = δhdc + δhdr
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(48)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat conduction through crack In the model the crack opening and crack sliding are applied
hdc = −κ
[[Θ]]s wn
(49)
where κ is the conductivity parameter of the medium between crack faces ( ) 1 1 [[Θ]]s =Θ+ + · ∇Θ+ · ws − Θ− − · ∇Θ− · ws = 2 2 (50) [[Θ]] + ⟨∇Θ⟩· ws The sliding part of crack opening vector ws can be expressed by crack multiplication of sliding direction sd and the value of crack sliding ws ⟨ ⟩ [[Θ]]s = [[Θ]] + ∇Θ · sd ws (51) J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat conduction through crack The derivatives of hdc can now be evaluated ∂hdc κ =− ∂[[Θ]] wn ws ∂hdc = −κ sd ∂⟨∇Θ⟩ wn ∂hdc [[Θ]]s =κ 2 ∂wn wn c ∂hd ⟨∇Θ⟩· sd = −κ ∂ws wn ∂hdc κws =− ⟨∇Θ⟩ ∂sd wn
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(52a) (52b) (52c) (52d) (52e)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat conduction through crack When taking into account evaluated derivatives in (52) then the derivative of hdc in relation to crack opening vector [[u]] reads as: ∂hdc [[Θ]]s κ κ = nd κ 2 − (sd ⊗ sd ) · ⟨∇Θ⟩ − (rd ⊗ rd ) · ⟨∇Θ⟩ ∂[[u]] wn wn wn
(53)
In the 3D case the following relation is satisfied: nd ⊗ nd + sd ⊗ sd + rd ⊗ rd = I
(54)
When taking into account relation from (54) to (53) it goes to the following form ) ∂hdc [[Θ]]s κ ( = nd κ 2 + · nd ⊗ nd − I · ⟨∇Θ⟩ ∂[[u]] wn wn
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(55)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat conduction through crack Remembering that δ⟨∇Θ⟩ = ⟨∇ δΘ⟩, then the corrective increment of hdc from equation (29) can now be expressed as follows ( ) ) [[Θ]]s ( κ κ c δhd = nd + nd ⊗ nd − I ⟨∇Θ⟩ · δ[[u]] − δ[[Θ]]s wn wn wn ws =const
(56) what finally can be expressed as: c c ∂h ∂h d δhdc = d · δ[[u]] + δ[[Θ]]s ∂[[u]] ∂[[Θ]]s
ws =const
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(57)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat conduction through crack
The κ parameter has the interpretation of the crack conductivity . It is the combination of air conductivity and bridging elements of the material. It is assumed that when the crack opens, at the beginning of the process some element of the materials bridges the crack faces. It can be written in the following way κ = (1 − d)κm + dκa where κm is the thermal conductivity of the material, κa thermal conductivity of the air, d is a crack damage parameter.
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(58)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat radiation through crack The model of the radiation crack heat flux hdr is based on Stefan-Boltzmann law. It is supposed in the paper that the amount of heat radiated out by hotter crack face is absorbed by the other crack face. In that case the radiation heat flux can be written as ( ) hdr = eσ · (Ts+ )4 − (Ts− )4 (59) Ts+ and Ts− – are absolute temperatures at both sides of crack regarding crack sliding, e – is the emissivity parameter 0 < e ≤ 1, σ – is Stefan’s constant (σ = 5.678 · 10−8 [W/m2 K]). The radiation heat flux can be expressed as follows: ( ) ( ) ( ) (60) hdr = eσ · (Ts+ )2 + (Ts− )2 · Ts+ + Ts− · Ts+ − Ts− and
( ) ( ) hdr = eσ · (Ts+ )2 + (Ts− )2 · Ts+ + Ts− · [[Θ]]s J. Jaśkowiec, L-5
Thermo-cohesive crack model
(61)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat radiation through crack
The following function of absolute temperatures on both sides of crack faces can be defined here ( ) ( ) H r = −eσ · (Ts+ )2 + (Ts− )2 · Ts+ + Ts− (62) then the heat flow through radiation is expressed by the relation similar to convecitve interface law hdr = −H r [[Θ]]s
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(63)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat radiation through crack The change of hdr is connected with changes of temperatures differences on both sides of the crack. The change of temperature jump is due to change of [[Θ]] and change of crack sliding ws . It can be written as ∂hdr ∂hdr r δhd = δ[[u]] + δ[[Θ]]s (64) ∂[[u]] ∂[[Θ]]s ws =const
( ∂[[Θ]] ∂hdr ∂hdr ∂[[Θ]]s 1 ∂[[Θ]]s ) s = = −H r · sd + · (rd ⊗ rd ) ∂[[u]] ∂[[Θ]]s ∂[[u]] ∂ws ws ∂sd (65)
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat radiation through crack Taking into account the definition of [[u]]s in (51) it can be easily written that ∂[[Θ]]s = ⟨∇Θ⟩· sd , ∂ws
∂[[Θ]]s = ⟨∇Θ⟩ws ∂sd
(66)
After incorporating (66) to (65) it result in ( ) ∂hdr = −H r · sd ⊗ sd + rd ⊗ rd ⟨∇Θ⟩ ∂[[u]]
(67)
Finally, after using the relation (54), the increment of radiation crack heat flux has the form ( ) r r r (68) δhd = −H · I − nd ⊗ nd ⟨∇Θ⟩ δ[[u]] − H δ[[Θ]]s ws =const
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Heat flow through crack ( hd = −
κ + Hr wn
) (69)
[[Θ]]s
The change of crack heat flux is due to change of crack opening and change of temperature jump between both crack faces ∂hd ∂hd δhd = δ[[u]] + δ[[Θ]]s (70) ∂[[u]] ∂[[Θ]]s ws =const
The derivatives in (70) can now be written as ( ) ∂hd κ =− + Hr ∂[[Θ]]s wn ) ( ( ) [[Θ]]s ∂hd κ = nd κ 2 + + H r · nd ⊗ nd − I ⟨∇Θ⟩ ∂[[u]] wn wn J. Jaśkowiec, L-5
Thermo-cohesive crack model
(71) (72)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
The modified Xu-Needleman cohesion law is used ( ) ( ) GI ws 2 wn wn tn = F · exp − exp − wnc wnc wnc wsc 2 ( ) ( ) GIIF ws ( wn ) ws 2 wn ts = 2 · · 1− exp − exp − wsc wsc wnc wsc 2 wnc
(73)
where GIF tnc exp(1) GII = √ F c ts 12 exp(1)
wnc = wsc
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(74)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
It is assumed that the cohesion law parameters: tnc , tsc , GIF , GIIF depend on temperature ( ) tn = tn wn , ws , tnc (Θd ), GIF (Θd ), tsc (Θd ), GIIF (Θd ) ( ) (75) ts = ts wn , ws , tsc (Θd ), GIIF (Θd ), tnc (Θd ), GIF (Θd ) In result the derivatives from (33) can now be written using the chain rule ∂tn ∂tn ∂t c ∂tn ∂GIF ∂tn ∂t c ∂tn ∂GIIF = c n + + c s + ∂Θd ∂tn ∂Θd ∂GIF ∂Θd ∂ts ∂Θd ∂GIIF ∂Θd ∂ts ∂tsc ∂ts ∂GIIF ∂ts ∂tnc ∂ts ∂GIF ∂ts = c + + c + ∂Θd ∂ts ∂Θd ∂GIIF ∂Θd ∂tn ∂Θd ∂GIF ∂Θd
J. Jaśkowiec, L-5
Thermo-cohesive crack model
(76)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
∂tn ∂wnc ∂tn = ∂tnc ∂wnc ∂tnc ( w ) ( w2) GI wn − 2wnc n s = − cF wn · exp − exp − 3 tn wnc wnc wsc 2 ∂tn ∂tn ∂tn ∂wnc = + ∂GIF ∂GIF ∂wnc ∂GIF wnc =const
( w ) ( w2) wn − wnc n s = wn exp − exp − 3 wnc wnc wsc 2 ∂tn ∂tn ∂wsc = ∂GIIF ∂wsc ∂GIIF ( ) ) ( wn wn 2ws 2 GIF ws 2 · exp − = · exp − wsc 2 GIIF wnc wnc wnc wsc 2 ∂tn ∂tn ∂wsc = ∂tsc ∂wsc ∂tsc ( ) ( ) 2ws 2 wn wn GIF ws 2 =− · · exp − exp − wsc 2 tsc wnc wnc wnc wsc 2 J. Jaśkowiec, L-5
Thermo-cohesive crack model
(77)
(78)
(79)
(80)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
∂ts ∂wsc ∂ts = ∂tsc ∂wsc ∂tsc ( ) ( ) (81) GII wsc 2 − ws ( wn ) ws 2 wn = 4 cF ws · 1 − exp − exp − ts wsc 4 wnc wsc 2 wnc ∂ts ∂ts ∂ts ∂wsc = + ∂GIIF ∂GIIF ∂wsc ∂GIIF wsc =const ( ) ( ) wn ) ws 2 wn ws wsc − 2ws 2 ( · 1 − exp − exp − = −2 wsc 3 wnc wsc 2 wnc (82) ∂ts ∂ts ∂wnc = ∂tnc ∂wnc ∂tnc ( ) ( ) GIIF wnc ws 2wn wnc − wn 2 ws 2 wn = −2 c · · exp − exp − tn wsc wsc wnc 3 wsc 2 wnc (83) ∂ts ∂ts ∂wnc = ∂GIF ∂wnc ∂GIF J. Jaśkowiec, L-5 Thermo-cohesive ( crack model ) ( )
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
XFEM interpolation In order to incorporate discontinuities in the displacements and temperature fields, XFEM approximation us used. The displacement vector and temperature field are approximated in a standard way: δu = Φu δˇ u,
ˇ δΘ = ΦΘ δ Θ
(85)
where Φu and ΦΘ are the approximation matrices for displacement and temperature fields respectively. The jumps of displacement vector and temperature are approximated by their approximations jumps: δ[[u]] = [[Φu ]] δˇ u
ˇ δ[[Θ]] = [[ΦΘ ]] δ Θ
(86)
The test functions, i.e. vu , vΘ , and their jumps are approximated in the same manner. The temperature gradient temperature approximation ˇ = BΘ δ Θ ˇ ∇ δΘ = ∇ΦΘ δ Θ J. Jaśkowiec, L-5
Thermo-cohesive crack model
(87)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
XFEM interpolation The crack temperature mid-gradient can be approximated in simmilar manner ˇ ⟨∇ δΘ⟩ = ⟨BΘ ⟩ δ Θ
(88)
( ) − where ⟨BΘ ⟩ = 0.5 B+ Θ + BΘ . The temperature jump increment in situation when the crack sliding is taken into account can now be approximated in the following way: ( ) ˇ = [[ΦΘ ]]s δ Θ ˇ δ[[Θ]]s = [[ΦΘ ]] + wsT ⟨BΘ ⟩ δ Θ (89) ws =const
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Linearized system of equations [
K + Kd Hu
KΘ + KΘ d H + Hd
][
] [ t+∆t ] Rf int,i δˇ u ˇ = Rh t+∆t δΘ int,i
where t+∆t t+∆t Rf t+∆t − fint,i int,i = fext t+∆t t+∆t Rh t+∆t int,i = hext − hint,i
∫
∫ BT u EBu dΩ ,
K= Ω
Γd
∫ BT u e ΦΘ dΩ ,
KΘ =
[[Φu ]]T
Kd =
∫ [[Φu ]]T
KΘd =
Ω
∂td [[Φu ]] dΓ , ∂[[u]] ∂td ⟨Φd ⟩ dΓ , ∂Θd
Γd J. Jaśkowiec, L-5
Thermo-cohesive crack model
(90)
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
Linearized system of equations (
∫ Hu = −
[[ΦΘ ]]T
∂hd ∂[[u]]
)T
(
∫ Hd = −
[[Φu ]] dΩ ,
Ω
[[ΦΘ ]]T
∂hd ∂[[Θ]]
) [[ΦΘ ]] dΓ ,
Γd
1 H= ∆t
∫
∫ ΦT Θ cρΦΘ
Ω
Ω
∫
∫ t+∆t fint,i
t+∆t BT u σi
=
Γd
∫
∫ ΦT Θ cρ
t+∆t BT dΩ + Θ qi Ω
dΓ , [[Φu ]]T td t+∆t i
dΩ +
Ω
t+∆t hint,i =−
BT Θ ΛBΘ dΩ ,
dΩ +
Ω J. Jaśkowiec, L-5
∆Θi dΩ − ∆t
∫ [[ΦΘ ]]hd t+∆t dΓ i Γd
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
The mesh updating Let us suppose that there are two interpolation matrices connected with two different meshes: Φ1Θ and Φ2Θ . The temperature field can be approximated by either of the approximation matrices: ˇ 1 ≈ Φ2Θ Θ ˇ2 Θ ≈ Φ1Θ Θ
(91)
ˇ 1 is known vector of degrees of freedom for the first mesh and where Θ ˇ Θ2 is the vector of degrees of freedom for the updated mesh that have to be calculated. An error between those two approximations is defined in the following way ∫ ( 1 ) ˇ 1 − Φ2Θ Θ ˇ 2 2 dΩ ΦΘ Θ ϵΘ = (92) Ω
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Modeling of heat flow through crack Modeling of cohesion tractions XFEM discretisation Mesh updating
The mesh updating ˇ 2 have to be chosen so that to minimizes the error that The vector Θ leads to the equation: ∫ ) ∂ϵΘ T( ˇ 1 − Φ2Θ Θ ˇ 2 dΩ = 0 =0 ⇒ Φ2Θ Φ1Θ Θ (93) ˇ ∂ Θ2 Ω
Eventually the vector of the degrees of freedom for the updated mesh is obtained by solving the equation ˇ 2 = M21 Θ ˇ1 M2 Θ
(94)
where M2 is the mass matrix for the updated mesh and M21 is the mass matrix based on the two meshes: the previous mesh and the current mesh. J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
Material parameters Property Density Poisson’s ratio Modulus of elasticity Coeff. of ther. expansion Specific heat capacity Thermal conductivity Reference temperature Tensile strength Fracture energy for mode I Shear strength Fracture energy for mode II emissivity parameter
Symbol ρ ν E α c λΘ T0 tnc GIF tsc GIIF e
Value 2300 0.2 27 14.5 · 10−6 0.75 1.8 20 3 0.08 3 0.08 2 · 10−3
Unit kg/m3 GPa 1/K kJ/(kg K) W/(m K) [o C] MPa kJ/m2 MPa kJ/m2
Table: Material parameters for the examples. The values are given for the reference temperature
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
Example 1 F
F
qn qn = 5 [kW/mm2 ] ˆ = 0 deg Θ
10cm ˆ Θ
3cm 8cm Figure: The geometry and boundary conditions for example 1 J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
Example 1
Figure: Temperature distribution in deformed structure without crack sliding modeling
J. Jaśkowiec, L-5
Figure: Temperature distribution with crack sliding modeling
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
Example 2 F F 10cm
qn = 5 [kW/mm2 ]
qn ˆ Θ
ˆ = 0 deg Θ
3cm 8cm Figure: The geometry and boundary conditions for example 2 J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
(a) Temperature without radiation
(b) Heat flux without radiation
(c) Temperature with radiation
(d) Heat flux with radiation
Figure: Temperature and heat flux maps for the example 2 with 0.1mm CMOV J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
(a) Temperature without radiation
(b) Heat flux without radiation
(c) Temperature with radiation
(d) Heat flux with radiation
Figure: Temperature and heat flux maps for the example 2 with 0.5mm CMOV J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
(a) Temperature without radiation
(b) Heat flux without radiation
(c) Temperature with radiation
(d) Heat flux with radiation
Figure: Temperature and heat flux maps for the example 2 with 2mm CMOV J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
(a) Temperature without radiation
(b) Heat flux without radiation
(c) Temperature with radiation
(d) Heat flux with radiation
Figure: Temperature and heat flux maps for the example 3 with 2mm and 8mm CMOV J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
The crack growth example ( tnc (Θd ) = tnc T0 1 −
Θd ) 580[o C] Θd GIF (Θd ) = GIF T0 (1 + ) 400[o C] ( 3 Θd ) · tsc (Θd ) = tsc T0 1 − 10 400[o C] Θd GIIF (Θd ) = GIIF T0 (1 + ) 400[o C]
(95)
In the continuum the Young modulus E and thermal conductivity λΘ parameters also change with temperature in the following way Θ E (Θ) = ET0 (1 − ) 600[o C] (96) Θ 1 λΘ (Θ) = λΘ T0 (1 − · ) 2 800[o C] J. Jaśkowiec, L-5 Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Material parameters Example 1 Example 2 Example 3 Example 4
Example 2 F F 10cm
qn = 5 [kW/mm2 ]
qn ˆ Θ
ˆ = 0 deg Θ
3cm 8cm Figure: The geometry and boundary conditions for example 4 J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
(a) Thermo-mechanical model
Material parameters Example 1 Example 2 Example 3 Example 4
(b) Mechanical model
Figure: The configuration in the thermo-mechanical analysis after 2.3 s
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
(a) Thermo-mechanical model
Material parameters Example 1 Example 2 Example 3 Example 4
(b) Mechanical model
Figure: The configuration in the thermo-mechanical analysis after 5 s
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
(a) Thermo-mechanical model
Material parameters Example 1 Example 2 Example 3 Example 4
(b) Mechanical model
Figure: The configuration in the thermo-mechanical analysis after 13 s
J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References
Conclusions The existence of the crack in the body has an influence on heat flow and temperature distribution The 3D model of heat flow through the crack has been presented that takes into account crack opening, crack sliding, radiation and bridging The model is described in local 2D subspace (nd , sd ), i.e. crack normal and crack sliding directions On feedback the temperature distribution has an influence on cohesion forces, crack path and crack growth criteria When the crack growths the mesh configuration is changing and the number of degrees of freedom is changing. In the incremental–iterative procedure the mesh updating is necessary The further effort is needed for full 3D example. J. Jaśkowiec, L-5
Thermo-cohesive crack model
Introduction Mathematical model Model of the thermo-cohesion crack Examples for crack heat transport Conclusions References J. Jaśkowiec. Three-dimensional analysis of a cohesive crack coupled with heat flux through the crack. Advances in Engineering Software, page 0, 2015. - accepted for publication. J. Jaśkowiec. Modelling of heat flow through a three-dimensional crack in thermoelasticity. In B.H.V. Topping and P. Iványi, editors, Proceedings of the Fourteenth International Conference on Civil, Structural and Environmental Engineering Computing, Stirlingshire, UK, September paper 65, 2014. doi:10.4203/ccp.105.65. Civil-Comp Press. J. Jaśkowiec. A coupled thermo-mechanical cohesive crack model in three-dimensional crack growth analysis. In B.H.V. Topping and P. Iványi, editors, Proceedings of the Fourteenth International Conference on Civil, Structural and Environmental Engineering Computing, Stirlingshire, UK, September paper 67, 2014. doi:10.4203/ccp.105.67. Civil-Comp Press. J. Jaśkowiec. Three-dimensional analysis of cohesive crack growth coupled with nonlinear thermoelasticity’. In B.H.V. Topping and P. Iványi, editors, Proceedings of the Fourteenth International Conference on Civil, Structural and Environmental Engineering Computing, Stirlingshire, UK, September paper 95, 2013. doi:10.4203/ccp.102.95. Civil-Comp Press. J. Jaśkowiec and F. P. van der Meer. A consistent iterative scheme for 2D and 3D cohesive crack analysis in XFEM. Computers & Structures, 136:98–107, 2014.
J. Jaśkowiec, L-5
Thermo-cohesive crack model
