Transport in Fracture Processes: Fragmentation and Slip
Roman TEISSEYRE and Marek GÓRSKI
Institute of Geophysics, Polish Academy of Sciences, Warszawa, Poland e-mail: [email protected]
A b s t r a c t
We present a new development in the asymmetric continuum the-ory with the shear oscillations (twist motions) and independent spin; these motions (displacement velocities and spin) can be shifted in phase to describe the independent rebound processes. Our approach provides an extension of the asymmetric continuum theory by including the micro-fragmentation processes with a double transport which may appear in an advanced fracture process under very high load. The related nonlinear equations, leading to soliton solutions, are derived.
Key words: asymmetric stresses, phase indexes, stress-defect relations, microvortex structure, rotation related transport, nonlinear transport processes, elastic string waves.
1. INTRODUCTION In this paper we consider the rotation field as a most important factor in the extreme and fracture processes. However, we admit that also in some inte-raction problems the rotation counterpart should be included. Some differ-ences were noticed between experiments and theory in problems including the electric and mechanical coupling. Roux and Guyon (1985) in order to improve the theory have proposed to include in the Hamiltonian the four contributions to elastic energy: extension-compression, flexion torque, strain shear, and torsion torque (cf., Crandall et al. 1978, Feng and Sen 1984, Feng et al. 1984, and Feng 1985).
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To understand fracture processes we shall first consider the microslip and micro-fragmentation processes; when approaching the microfracture and fracture states we shall consider consecutive substantial changes in the ma-terial properties and governing equations. The material properties undergo changes, e.g., from the elastic to plastic and, further on, to crushed, granu-lated and even partly melted state (mylonite). Moreover, an influence of the rotation processes of different nature and scale may be of great importance when some vortex microstructures will appear. In any fracture process (ex-cept strong explosions) the molecules elastically rotate and precede the breaks of the molecular bonds with rebound rotation movements.
2. LINEAR CONSTITUTIVE RELATIONS We follow the asymmetric continuum theory developed in our former papers (Teisseyre and Górski 2007, 2008, 2009, Teisseyre et al. 2008, Teisseyre 2009) with the asymmetric stresses, strain and rotation.
We start with the perfect elastic condition in a frame of the antisymme-tric continuum theory; for the asymmetric stresses Skl , symmetric strains Ekl , and antisymmetric rotations ωkl , we write the following equations:
( ) [ ] ( ) [ ], , ,ω ω= + = =kl kl kl kl kl kl klS S S E E
( ) [ ]2 , 2 , (3 2 ) ,μ μ ω λ μ= = = +D Dkl kl kl kl ss ssS E S S E
0 0 0 01 , ,2kl kl l k kl kl l k
k l k l
E E u u u ux x x x
ω χ ω χ⎛ ⎞ ⎡ ⎤⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦
∂ ∂ ∂ ∂= = + = = −
∂ ∂ ∂ ∂ (1)
where DklS and D
klE mean the deviatoric parts of tensors, and the phase index constant 0χ may take the following values: 0 {0, 1, i}.χ = ± ±
In this approach we assume that fracture processes may generate inde-pendent slip and rotation fields; however, at a common microfracture event these fields may be related to each other with the help of the introduced phase shift constant 0;χ therefore, in relations (1) there appears the constant
0χ in the expression for rotation.
3. FRACTURE PROCESSES In fracture process, some independent motions are generated; however, the displacement and rotations may be simultaneously related with a possible phase shift 0χ . For simplicity we assume that during the fracture process a compressibility remains practically unchanged. We can easily include the progressive changes into the constitutive laws for the deviatoric parts of the
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stress tensor and for its antisymmetric part; e.g., we may consider the follow-ing linear constitutive laws:
( ) ( ) [ ] [ ]2 2 , 2 2 ,τ μ η τ μω ηω+ = + + = +D D D Dik ik ik ik ik ik ik ikS S E E S S (2a)
where besides the rigidity μ, there appear the relaxation time τ and viscosity η. At the final stage of the crushed and granulated rock (pretty close to fluid
properties), there remain only the time rates of the strain ( ) [ ]2 , 2 ,τ η τ ηω= =D D
ikik ik ikS E S (2b)
where ( )DikS and [ ]ikS denote the deviatoric and antisymmetric parts of stresses.
For these fields we may also introduce the structural phase indexes (Teisseyre 2008b, 2009); this approach will permit an independent genera-tion the displacement and rotation motions with the appropriate phase shifts
0 0 0 1 ,2
l kik lk
k l
u uE e E ex x
⎛ ⎞∂ ∂= = +⎜ ⎟∂ ∂⎝ ⎠
(3a)
0 00 1 ,2
l kkl kl
k l
u ux x
ω ωχ χ⎡ ⎤∂ ∂
= = −⎢ ⎥∂ ∂⎣ ⎦ (3b)
where the coefficients e0 and 0χ are the phase constants, which may differ from those in eq. (1).
Under a shear load, the micro-fracture processes can run as follows: shear stresses and related strains cause some changes in the angular molecule orientations, then the slip motion and break of the molecular bonds starts with an immediate drop of shears, which is followed by the rebound rotation retarded in phase.
Under compression load, the induced defects produce centers with oppo-site shears which may be combined with rotation, some microbreaks lead to the rotations and fragmentation process, which is followed by the rebound slip motions retarded in phase.
In the first case, the shears create the dynamic angular deformations leading to the bond breaks and slip propagation followed by the rebound ro-tations retarded in phase. In the second case, the microfractures under com-pression lead to the opposite sense of the induced shear motions: the twist motions and the related fragmentation and granulation processes precede the slip rebounds retarded in phase.
4. MOTION EQUATIONS We will repeat here some basic elements from our former studies (Teisseyre and Górski 2007, 2008; Teisseyre 2009).
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For the elastic asymmetric continuum theory we can consider the follow-ing constitutive relations (separately for the deviatoric fields ( )
DklS and D
klE ):
( ) [ ] ( )2 , 2 2 .λδ μ μω μ= + = =D Dkl kl ss kl kl kl kl klS E E S S E (4)
The motion equation 2 2( ) / / /kl k l l lS x u t F p xρ∂ ∂ = ∂ ∂ + −∂ ∂ (containing
pressure gradient) leads to
2 2 2 2 2
2
1 2
n lss nl ss nl
n l k k l n l n n l
F F pE E E Ex x x x x x t x x x x
λ μ ρ⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂ ∂ ∂ ∂
+ + = + + −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠
and we arrive at the wave equations for the axial and deviatoric parts
( )2 2 2
22 ,λ μ ρ∂ ∂ ∂+ = −
∂ ∂ ∂ ∂∂ss ssk k s s
E E px x x xt
(5)
2 2 2 2
2 ( ) + ,3δ
μ ρ λ μ⎛ ⎞∂ ∂ ∂ ∂
− = − + −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂∂ ⎝ ⎠D D nlnl nl ss ss nl
k k l n k k
E E E E Yx x x x x xt
(6)
where 2 21 1
2 3⎛ ⎞∂ ∂ ∂ ∂
= + − +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠nl n l
l n s s n l
Y F F p px x x x x x
at 0∂
=∂
n
n
Fx
.
For the antisymmetric stresses [ ] ,niS and the couple of forces [ ] ,kiK we can take one of the following equivalent relations (with the compatibility:
2 / 0imk jns ks m nx xε ε ω∂ ∂ ∂ = and / 0; s s s skn knxω ω ε ω∂ ∂ = = ):
22
2 2 2[ ] 2 2 ,ni i
pk pki ni pki pkik k n k n k
M l S l lx x x x x x
ω Ωε με με∂ ∂∂ ∂= = =
∂ ∂ ∂ ∂ ∂ ∂ (7a)
where ,ω
Ω∂
=∂
nii
nx or
2
[ ][ ] [ ]2 2 ,ni
pki pki ki pki ki pki kik n
SS K
x xε ε ε ρω ε
∂= Δ = +
∂ ∂ (7b)
where [ ]ki ki kiKμ ω ρωΔ = + ; the angular rotation moment is given as 2
nl nl xω∂ ∂ and l is the characteristic Cosserat length. These relations replace those for the stress moment balance
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2
[ ][ ]2
1 2 2 ,kp nipki pki ki pki ki
k k n
M SK
l x x xε ε ρω ε
∂ ∂= = +
∂ ∂ ∂
(7c)
2 22 2 .nlkp pki pki l
n
M l lxωε μ ε μ Ω∂
= =∂
Relation (6) for deviatoric stresses DikE may be transformed to its off-
diagonal form for the shear-twist pseudovector, 23 31 12{ } { , , },= D D DsE E E E and
remains valid for any system with the help of the 4D Dirac tensors γ (Teis-seyre 2009)
3 2 1
1 2 3 13 21 2 3
32 1
1 2 3
00
i i .0
0
E E EE E E
E E E EE E EE E E
λκ γ γ γ
⎡ ⎤− −⎢ ⎥− −⎢ ⎥= + + =⎢ ⎥− −⎢ ⎥⎣ ⎦
(8)
Finally, we obtain the wave form
2
2 ,λκ λκ λκμ ρ ∂Δ − =
∂E E Z
t (9)
where
2
( ) + ,λ μ ∂= − +
∂ ∂lk ss lkl k
Z E Yx x
2 21 1 ,
2 3lk k ll k s s k l
Y F F p px x x x x x
⎛ ⎞∂ ∂ ∂ ∂= + − +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
and 1 2 323 31 12i iZZ Z Zλκ γ γ γ= + + . The shear-twist, 1
2s slk lkE Eε= , is defined
as the off-diagonal oscillation of shear axes and its amplitude. We arrive at the 4D conservation law for the rotation tensor with the spin
and twist motions
1 2 3 4 44πi , , { , , , }, i ,E J x x x x x x Vtx V
λλκ λκ λκ λκ λκω ω ω∂= + = = =
∂ (10)
and further we have the following relations for the vectors, ,ω ε ω=s skn kn 12
,s skn knE Eε= (Teisseyre 2008b, 2009):
0 0rot 4π J , rot 0,EV V Et t
ωω ∂ ∂− = + =∂ ∂
(11)
where V0 is the wave propagation velocity.
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Interrelation between the shear-twist and spin may be associated with shear twirl oscillation and whirl processes.
In particular case we may recall the local solution of the system (11) for the shear-twist and spin waves (partial derivatives) shifted in phases by π/2 (Teisseyre et al. 2008b)
[ ] [ ]0 0i exp i( ) , exp i( ) ,s s s s i i s s i iE k x t E E k x tω ω ω ω ω= : = − = − (12)
where we demand 0 0is sEω = and for the constants:
0 0 exp(i ) ,s s sω ω ψ= 0 0 exp(i ).s s sE E ϕ=
With the waves introduced, is sEω = , we arrive at a possibility to study the dynamic objects and to explain the synchronization of the microfracture phenomena.
5. TRANSPORT EQUATIONS: SLIP For advance deformation state we believe that not only the constitutive laws shall be progressively changed but that some essential changes should be introduced into the motion equations as well. In fluids we deal with the Navier–Stokes transport equation; here, in solids under fracture process, it seems necessary to include both the slip-transport and rotation-transport, especially when some vortex microfracture structure related to micro-fragmentation of material start to appear.
For simplicity we can assume that during the fracture processes a materi-al density remains constant; therefore
2 2
2 .i i ik k i
u u p Ft x x x
ρ μ∂ ∂ ∂= + −
∂ ∂ ∂ ∂ (13)
We shall note that for the fluids under stationary conditions (without a transport process) we have
2
i i ik k i
p Ft x x x
ρ υ η υ∂ ∂ ∂= + −
∂ ∂ ∂ ∂ ,
where we had to exchange: i iuμ ηυ→ , while the related Navier–Stokes transport equation becomes
2d .
dρ υ η υ∂ ∂
= + −∂ ∂ ∂i i i
k k i
p Ft x x x
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For solids, when including a transport process, we put
2 2
2 2
d ,d
υ υ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂
→ = + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂∂ ⎝ ⎠⎝ ⎠i i s i s i
s s
u u u ut x t xt t
(14)
where /u tυ = ∂ ∂ . Instead of eq. (13), we write, in a similar manner, the Navier–Stokes
transport-fracture equation under the microfracture processes
2 2
2
2
2
1 .
i i i s ik k k k s i
k k k s k s
i i
i k k
u u u ut x x x x x x
u Fpx x x
υ υυ υ υ υ υ
μρ ρ ρ
∂ ∂ ∂ ∂ ∂ ∂+ + + +
∂ ∂ ∂ ∂ ∂ ∂ ∂
∂∂= + −
∂ ∂ ∂
Moreover, we include a transport process for the balance of stress mo-ment (eq. 7b):
2 2
[ ] [ ]2 2
dd
ρ ω μ ω ρ ω μ ω∂= Δ − → = Δ −
∂ ki ki ki ki ki kiK Kt t
and we obtain in a similar way
2 2
[ ]2 2 .ω ω ω υ ω ω μυ υ υ υ υ ω
ρ ρ∂ ∂ ∂ ∂ ∂ ∂
+ + + + = Δ −∂ ∂ ∂ ∂ ∂ ∂∂
nini ni ni s ni nik k k k s ni
k k k s k s
Kx x x x x xt
(16)
The obtained relation describes transport in a slip process.
6. DOUBLE TRANSPORT PROCESS: SLIP AND FRAGMENTATION Acting with the curl operator on the Navier–Stokes transport-fracture equa-tion (15), we can arrive at the relation for the classical vorticity ς
2
2 2 2ς υ ς υ υ ς υ υ
ε υ ε υ ε∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂
n k i n k i n k s inpi k npi k npi
p k k p k k p k s
u ux x x x x x x x xt
2 2 2υ υ ς υ υ
ε υ υ ε υ ε υ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
s i s n k i s inpi k k npi s npi k
p k s k s p k s p k s
u u ux x x x x x x x x x x
22 21 1 ,n i
k s n npi npik s p i k k p
Fpx x x x x x x
ςς μυ υ ς ε ερ ρ ρ
∂ ∂∂ ∂+ = + −
∂ ∂ ∂ ∂ ∂ ∂ ∂
where /n npi i pu xς ε= ∂ ∂ . Further, we may introduce the criterion justifying the double transport
process: when the ratio of the absolute value of vorticity to rigidity,
(15)
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abs( )/ς μ , overpasses some given value, then the rotation counterpart should be included.
Thus, according to definitions in (7a) we can present expressions for the double transport process
( )2 2i i i i
d, , ,d
nii s s
n s
l lx t t xωυ υ υ υ ρ υ υ∂ ∂ ∂
= + = = Ω → + +∂ ∂ ∂
(17)
where in such a double transport theory we consider both the displacement
velocity, /u tυ = ∂ ∂ , and the spin related transport velocity, 2 2nii
In this way we arrive at a description of the vortex-fracture structures. The displacement transport term, υ , is most important in the vicinity of
slip motion under the confining shear stresses; under the compression load the displacement transport is less important, while a major role will be played by the rotation related transport υ .
Thus, at a constant density the double transport equation in solids subject to shears or fragmentation can be written according to the above expression and to eqs. (14) and (17) as follows:
22 2
2 2 2υ υ υ
υ υ Ω Ω υ∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + + + +∂ ∂ ∂ ∂ ∂ ∂∂
i i i i i s ik k k k k
k k k k k s
u u u ul l
x x x x x xt
2 2 2
2 22Ω υ
υ υ υ υ Ω Ω∂ ∂ ∂ ∂ ∂ ∂
+ + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
i s i i s ik s k k s k
k s k s k s k s
u l u u ul l
x x x x x x x x
2 2 2
2 2 2 1 .Ω μΩ Ω Ω
ρ ρ ρ∂ ∂ ∂ ∂∂
+ + = + −∂ ∂ ∂ ∂ ∂ ∂ ∂
s i i i ik k s
k s k s i k k
l u u u Fpl l lx x x x x x x
(18)
Moreover, we include a transport process to the balance of stress mo-ment (eq. 7b):
2 2
[ ] [ ]2 2
d .dki ki ki ki ki kiK K
t tρ ω μ ω ρ ω μ ω∂
= Δ − → = Δ −∂
We obtain similarly to (18)
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22 2
2 2 2ω ω ω ω ω υ ω
υ υ Ω Ω υ∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + + + +∂ ∂ ∂ ∂ ∂ ∂∂
ni ni ni ni ni s nik k k k k
k k k k k s
l lx x x x x xt
2 2 2
2 22ω Ω ω ω υ ω
υ υ υ υ Ω Ω∂ ∂ ∂ ∂ ∂ ∂
+ + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
ni s ni ni s nik s k k s k
k s k s k s k s
ll l
x x x x x x x x
2 2
[ ]2 2 2 .nis ni nik k s ni
k s k s
Kll l lx x x xΩ ω ω μΩ Ω Ω ω
ρ ρ∂ ∂ ∂
+ + = Δ −∂ ∂ ∂ ∂
(19)
The obtained relations describe transport in slip and fragmentation pro- cesses.
7. ROTATION PROCESSES IN THE TRANSPORT PHENOMENA Equivalently to relations (18) and (19) we can apply the generalization of the Navier–Stokes transport idea to the combined rotation and shear-twist fields, ωk and kE (eqs. (8) and (10); cf., Teisseyre 2008b, 2009; Teisseyre and Górski 2008, 2009) we obtain:
shear-twist and rotation in the system of the Maxwell-like equations:
0 0d drot 4π J, rot 0,d dEV V Et t
ωω − = + = (20a)
shear-twist and rotation in the wave related equations:
2 2
2 2 2 2 20 0 0 0
d 4π d 4π, 4π ,d dnpq q n
p n
J E E JV t V x V t V x
αω ω ∂ ∂Δ − = − ∈ Δ − = +
∂ ∂ (20b)
where sE and ωs present the shear-twist and rotation vectors (cf., eq. (11)), and where instead of the couple forces, [ ]niK , we write the current related defect flow, e.g., dislocations and parameter α related to defect density.
Following our consideration on transport processes, we shall note that the rotation part may lead us to more complicated vortex structures under high external pressure load; the external transport field, u, could be treated further on as a given external function, while the internal transport field, u , will appear as additional terms introduced to the system of equations. This rotation transport may lead to the local vortex structure; in principle, a de-crease of density caused by the centrifugal motion should be included inside such a structure; however, for our further consideration on fragmentation in solid body we will keep density as a constant parameter.
According to (16) and (20) we may write for the double transport process
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20 0
d 0,d
n n n n n nnls k k nls
k k k k
E EV l Vt x t x x xω ω ω ωυ Ω∂ ∂ ∂ ∂ ∂
+ ∈ = + + + ∈ =∂ ∂ ∂ ∂ ∂
20 0 0
d 4πd
n n n n n nnls k k nls n
k k k k
E E E EV l V J Vt x t x x x
ω ωυ Ω∂ ∂ ∂ ∂ ∂+ ∈ = + + − ∈ = −
∂ ∂ ∂ ∂ ∂. (21)
However, it seems better to start with the related wave type forms (16) de-duced from the above set
2 2 22 2 2 2
0 02
0
dd
4π ,
n n n n n ns s k k
k k s s k k k k
npq qp
V l l Vt x x t x x t x x x x
V Jx
ω ω ω ω ω ωΩ υ Ω υ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂∂ ∂ ∂
− = + + + + −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠∂
= ∈∂
2 2 22 2 2 2
0 02
dd
n n n n n ns s k k
k k s s k k k k
E E E E E EV l l Vt x x t x x t x x x x
Ω υ Ω υ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂∂ ∂ ∂
− = + + + + −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠
204π 4π .n
n
J Vxα∂
= − −∂
(22)
We obtain the expression equivalent to (19):
2 22 2
02
2 2 22 2 2
2
2 2
pkn n n n n s nk k k
k p k k k k k s
pk pk pkn n n s nk s s s
k s p k s p k s p k s
l Vt x x x x x x x x
l l lx x x x x x x x x x x
ωω ω ω ω ω υ ωυ υ υ
ω ω ωω ω ω υ ωυ υ υ υ
∂∂ ∂ ∂ ∂ ∂ ∂ ∂+ + − + +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂∂ ∂ ∂ ∂ ∂+ + + +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
2 22
2 2 2 22pk pk pk qsn n ns
s p k p k p k q s
l l l lx x x x x t x x x xω ω ω ωω ω ωυ
∂ ∂ ∂ ∂∂ ∂ ∂+ + +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
2 2
2 204πpk qs n
npq qp q k s p
l l V Jx x x x xω ω ω∂ ∂ ∂ ∂
+ = ∈∂ ∂ ∂ ∂ ∂
(23)
and
2 2 22 2
02
2 2 22 2
2
2 2
pkn n n n n s nk k k
k p k k k k k s
pk pkn n nk s s s
k s p k s p k s
E E E E E El Vt x x x x x x t x x
E E El lx x x x x x x x
ω υυ υ υ
ω ωυ υ υ υ
∂∂ ∂ ∂ ∂ ∂ ∂ ∂+ + − + +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂∂ ∂ ∂+ + +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
2 2
2 2 22pk pk pks n n ns
p k s s p k p k
E E El l lx x x x x x x x tω ω ωυ υ∂ ∂ ∂∂ ∂ ∂ ∂
+ + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
2 2
2 2 2 2 204π 4π .pk qs pk qsn n
np k q s p q k s n
E El l l l J Vx x x x x x x x xω ω ω ω α∂ ∂ ∂ ∂∂ ∂ ∂
+ + = − −∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
(24)
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8. SHEAR AND CONFINING LOADS In a particular case with the time constant slip velocity along one direction only, 1 1 2( )u xυ = , and for 3 1 2( , , )x x tω ω= , 3 1 2( , , )E E x x t= , we obtain that a transport according to eqs. (23) and (24) becomes related only to 1 2( )xυ and
We have obtained the nonlinear (soliton) transport equation for the spin mo-tion; such a transport may lead to a local vortex microstructure.
First, we can consider the processes under a shear load; the double trans-port, displacement motions and spin generate the microvortex structures sub-jected simultaneously to displacement transport. In the material inside the slip zone, called mylonite, the microrotation processes appear at both sides of the inner fracture zone. However, in reality the rotations along the slip fragments perpendicular to the main slip plane will be opposite; hence, the resulting rotations would be almost compensated, as schematically shown in Fig. 1.
For the confining load, the related equations for the microvortex struc-ture can lead to fragmentation centers. Due to much lower shear strength, there appear induced centers with the opposite shear signs; shears become almost compensated, while the separate vortex centers may have different sense of rotations; this situation helps create separate vortices leading, fur-ther on, to the fracture fragmentation phenomena, as schematically shown in Fig. 2.
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594
Fig. 1. Sketch of slip elements and op-posite rotations.
Fig. 2. Fragmentation centers and in-duced opposite shears.
For both cases we obtain the nonlinear soliton equations for spin ω and linear type equation for shear-twist E . We can suppose that such a transport of rotation motion becomes possible only in the granulated structures or in those undergoing the micro-fragmentation processes.
9. ISOLATED VORTEX In further stage, for the inner vortex structure, especially important for the case of compression load, we can introduce the condition of conservation of the moment of momentum (constancy of rotation moment in the absence of an external torque) inside each isolated vortex structure. Such a vortex struc-ture reminds the structures in the micromorphic continua; also here, inner deformation processes take place in each vortex element.
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We consider an isolated vortex, {0,0, }zω ω= and 12zω ω= , with vanish-ing displacement velocity, 0υ = , in which we neglect the change of density due to the centrifugal forces. At the absence of external torque we can intro-duce the condition of conservation of the moment of momentum dM /dt = 0
d d2 0 , 2 const. ,d dM Ll M l Lt t
Ωμ μ Ω= = = =
1 2
or 2 2 ,ks
k
M l L l Lx x xω ω ωμ μ
⎛ ⎞∂ ∂ ∂= = −⎜ ⎟∂ ∂ ∂⎝ ⎠
(27)
where we have introduced the time variable arm L in the expression for mo-ment.
We obtain
1
1 22Ml L
x xω ω
μ
−⎛ ⎞∂ ∂
= −⎜ ⎟∂ ∂⎝ ⎠
and finally the nonliner spin relation (20b) becomes as follows:
We may note that such inner vortex processes can serve as a model of the vortex soliton phenomenon, which appears due to the non-linear structure of the derived equations.
We may note that there exists a similarity between these two sets of equ-ations related to rotation and twist motions. These two systems might be si-multaneously solvable for s skEω = ; we obtain the synchronized solution fitting to relation (12): is sEω = ± (Teisseyre 2008, 2009). We can conclude that the rotation fields, rotation and twist, shifted in phase by π/2 (such a phase shift may be caused by a fracture of some molecular bonds related to rotation motion which releases the retarded motion of twist, or vice versa) could undergo a transport process.
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10. FRAGMENTATIONS AND CRACKS APPROXIMATED BY MICRODEFECT ARRAYS
Following paper by Teisseyre 2008a (cf., Kossecka and DeWitt 1977) the Burgers and Frank vectors and dislocation and disclination densities are given as
d dl kl lqr kq r k kl kl kB E x l E lε χ ω⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦
= − = +∫ ∫
and d dq pq p pq pl sΦ χ θ= =∫ ∫∫ ,
where the introduced twist-bend tensor mqχ was defined as
12
m nsmq nsm
q qx xω ωχ ∈∂ ∂
= =∂ ∂
.
Some later improvement (Teisseyre 2001, Teisseyre and Boratyński 2002) was achieved with another definition
.mkmq ksq
sxωχ ∈ ∂
=∂
However, with the proposed definitions for deformation fields (1) or (3) we can define the combined Burgers and Frank vector as follows:
0 0 1d d d ,2l kl kl k kl kl k pl pl ss pB E l e E l sω χ ω α δ α⎡ ⎤⎡ ⎤
⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦
⎛ ⎞= + = + = −⎜ ⎟⎝ ⎠∫ ∫ ∫∫ (30)
where for e0 = 1 and χ0 = 0, symbol plα means the dislocation density, and for e0 = 0 and χ0 = 1 it means the disclination density θpl
.pl pl plα α θ= + (31)
These densities are related to strain, rotation and stresses in the following way:
( )1 1 1
2 2
kl kl sskl
pl pl ss pmk pmkm m
S SEx x
ν δνα δ α ∈ ∈
μ
⎛ ⎞∂ −⎜ ⎟∂ +⎝ ⎠− = =∂ ∂
, (32)
[ ]12
p klkl spl pmk pmk kls pmk
m m l m
Sx x x x
ωω ωθ ∈ ∈ ∈ − ∈
μ∂ ∂∂ ∂
= = = =∂ ∂ ∂ ∂
, (33)
where ν is the Poisson ratio. The dislocations can form arrays leading to cracks, while the disclination
arrays defined above (related to gradient of rotations) will form the micro-vortex defects or fragmentation-cracks.
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Fig. 3. Disclination line with the re-lated rotational deformation (eq. 33).
11. ELASTIC STRING WAVES An important role of the defect densities may appear in the problem of the coupled shear-twist and rotation waves; the conjugated phase shifted solu-tion (12) can be considered not only in the case of the near-fracture process but also in the case in which the elastic oscillation of dislocations will effi-ciently influence the rotation motions, e.g., due to an oscillating external shear condition. Such oscillations of dislocation field lead us to the elastic string waves appearing in an elastic domain, not exclusively when the micro-fracture processes occur. The elastic string wave relate to the shear-twist de-fect oscillations and to coupled rotations with the π/2 phase shift (eq. 12):
i .s sE ω= ± A next step along this line of consideration could lead us further to the
problem of fatigue cracks when the defect oscillation will be close to some resonance material strength frequency.
12. CONCLUSIONS Under the high compression loads there will appear some advanced changes in the material properties; when rotation motions and deformations become significant, some microfragmentation or granulation processes can lead us to formation of microvortex structures. When rotation motions and deforma-tions become significant, the transport process may involve additionally an action of rotation with the appropriate Cosserat length unit. Moreover, some microfragmentation or granulation processes can lead us to formation of the microvortex structures.
The complex transport motion, with displacement rotations, leads to the nonlinear motion equations. In particular, some soliton solutions might be
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searched. States with some micro-rotation structure formed by the disclina-tion defects have to be considered.
We have derived the relations between the defects, dislocations and dis-clinations, and the strain and rotation fields, or equivalently the symmetric and antisymmetric stresses.
We have indicated that under some oscillating stress load the elastic string wave related to the shear-twist defect oscillations and to coupled rota-tions may appear even in an elastic domain, completely not related to the ex-treme deformation conditions.
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