Thermal deformation, complex bending and stretching of lithospheric plate in tectonic force field

Автор: Baimukhametov A. A., Kultasov A. A., Abdimanapova P. B., Mazhit Zb. B.

Журнал: Вестник Алматинского технологического университета @vestnik-atu

Статья в выпуске: 2 (127), 2020 года.

Бесплатный доступ

Plate tectonics implies the rigidity of near-surface Earth rocks and their elastic behavior at geological scales of time. The plates are subjected to tectonic and thermogradient forces resulting in their bending, stretching and thermal deformation. The article considers the new setting of the problem of complex bending and stretching of the axisymmetric lithospheric plate of variable thickness caused by both force effects and uneven heating. The mathematically considered problem is reduced to a differential equation with variable coefficients, the analytical solution of which was first possible to obtain by the method of partial sampling.

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Lithospheric plate, complex bend, stretching, transverse forces, temperature field

Короткий адрес: https://sciup.org/140250885

IDR: 140250885

Текст научной статьи Thermal deformation, complex bending and stretching of lithospheric plate in tectonic force field

The plate tectonics model implies the stiffness of near-surface rocks and their elastic behavior at geological scales of time. The plates are subjected to tectonic and thermogradient forces resulting in their bending, stretching and thermal deformation. Thin resilient surface plates form a lithosphere which floats on the underlying relative liquid mantle. The plates experience various loads, such as the weight of volcanoes and seamounts, so that they bend. For the study of such geological phenomena, the theory of bending of plates under the influence of applied forces and moments of forces is applicable. With this theory it is also possible to explain the

occurrence of series of folds in mountain belts, as the formation of folds can be seen as deformations of elastic plates under the action of horizontal compressive forces. The theory of plate bending is used to simulate formation blasting over magmatic intrusions [1-3].

Objects and research methods

The problem of joint bending and stretching of non-uniform non-composite plate of variable thickness in non-uniform temperature field is considered. Partial sampling is used to find an analytical solution to the problem [4]. The basic differential equations of quasi-static equilibrium are reduced to the next second-order differential equation with respect to angular displacement

d 2 9 ( 1 1 dD_M ) d S ( v dD_M N 1 1

+ +- I 1 +MrIS dr I r DM dr ) dr I rD- dr DM r )

rD rM

⁽J q iz Kd ^_ c

1 + vd

DMdr

(Xtdm ) - 0

d N + 3_ J_dDN 1 dN- -F(1 -v)dD N dr2 (r DN dr J dr rDN dr 1

1 dq q rdrr

1 + v I

^dD N J ₊ 1 _ v l

DN dr

D^d ^T- — 0

N dr

<4 — 5W where - angular movement, dr

– deflection, DM – cylindrical rigidity of a bend, – Poisson's coefficient, r – position of midplane point before its deformation, – thermal deformation due to non-uniform heating,

– intensity of cross forces.

An edge task is set when a circular plate of variable thickness with a rigidly embedded inner contour is loaded along its outer contour

. Boundary conditions will take the form

( )

1 ^k j

X — Z A s r j , A s — const (3) Т h j — 0 j j

In addition, let the plate of variable thickness rigidly embedded on the inner office r — r be loaded uniformly distributed along the surface by transverse forces of intensity q and along the contour r — r2 by transverse force Q k qr — Z q jr3, qj — const (4) j—0

Then the general solution of equation (1) under an arbitrary law of change in plate thickness will be

Let the plate be subjected to uneven heating. In the case of linear heat propagation a T in the thickness of the plate the thermal deformation is approximated in the form of

9 — B + A J e J5(r)drd.r + Je J4(r)dr(J[n(r) + g(r) + ф(г)]е J^(r)d'dr)dr

where

П (r) = ^- v X ln

D N ⁽ r k ) ■ r k - 1 )

D_ON

^r k - 1

⁷ ⁽ ^r ^- ^r k - 1

) - In

d m ⁽ r k ) ■ r k )

D_ON

^r k + 1

5 ⁽ r ^- r k

)

_-

A

_-

X ¹ ■ ( r . - 1 ) 7 (

A r k 7

^+X ^ ⁽ ^r k

A

r

_-

^r k - 1 ⁾ ^-

1 ■ ( г . W

A r k 7

: r ^- rk ) +

^ ⁽ ^r ) =

_-

r/3

rD_M

^N r ⁽ r k - 1 )

^D M ( ^r k - 1 )

■7 ^r k - 1 7 1 ^r

_-

r k - 1 )

_-

^N r ⁽ r k )

D_M ( r_k )

■ ⁽ ^r k ⁾ ⁷ ⁽ ^r ^- ^r k - 1

)

( J q z rdr ^- C )

Ф⁽ ^r ) =

1 + v d

—

D M

dr

( X t d m ); ^ ⁽ ^r ) = ¹ ⁺ r

1 dD M

D M

dr

.

Arbitrary coefficients A and B

are

а о

-13 в

determined by boundary conditions (2).

The cylindrical bending stiffness is

d m = D 0

—

.

Using the boundary conditions (2) we get

■ ( r ) = 0.2

Mr ₀

-

+

q ₀ r ₀ ³

3D

OM

-

^D OM

0.346

q ₀ r ₀

-

1ln 1

-

A

J_x ( r ) v X ln

-

ln

r

A

+

^r o 7

^D OM

0.42

Qr 0²

D

OM A

0.43

M

-

r o 1

A

r

r

A

-

-

^r o 7

16 1

A

r

A

-

r o 7

^D OM

0.04

q ₀ r ₀

—

+

^D OM

Qr 0²

4 D OM

A

-

r

r o 7

0.236

Qr ₀

A

Г)

^OM 7

J 1 ⁽ ^r ^{) +}

-

d m ⁽ r k ) ■ r k - 1 )

^D OM

r ₀

A

-

rk - 1

^r o 7

^H ⁽ ^r ^- ^rk - 1

⁾⁽ ^r ^- ^rk - 1

)

-

d m ⁽ rk ) ■ r k )

Г

^D OM

A

-

r_k

r ₀

A

r ₀

H ( r

-

^rk - 1 ⁾⁽ ^r

—

r k - 1 ) ⁺

+ X f^ ■ ( r_k - 1 ) r

^rk - 1

A ^r k 7

r ₀

A

r o 7

H ( r

^rk - 1 ⁾⁽ r

r k - 1 )

¹ ^ ( r . ) r-

A ^r k 7

r ₀

A

r_k

r o 7

H ( r

r k ⁾⁽ ^r

^r k )

>

^X ^ ⁽ r k )

N r ( г . - 1 )

D M ( г . - 1 )

^ ( г . - 1 ) r k ^- ¹ 1

r ₀

A

^r k - 1

^r o 7

^H ⁽ ^r ^- ^r k - 1

⁾⁽ ^r ^- r k - 1

)

^“

N r ( r k )

D_M ( r_k )

r

■ ( r . ) - k- 1

r ₀

A

r_k

r ₀

^H(r ^- ^rk^)(r

^“

r_k)

^,

d S dr

0.43 M - 0.04 q_or_o ² - 0.236 Qr'

+

0.5 Qr ₀ ² - 0.125 q_or _o ² - 0.166 q_or _o ²

D o 1 -

V

r

r ₀

r   1      r I

¹ I

^r 0 V     ^r 0 J

v zT ln D M^ r k ) ( 1 -У H ( r - r ,) - In D^ ( 1 -T H ( r - r ,) D\       k 1 /                                              \       k 1 /

OM    ^r 0    V     ^r 0 J                   D OM    ^r 0 V    ^r 0 J

+ 2 ⁽1 I s ( r .- 1 ) r

_-

r k - 1

_-

+

V r k

^r 0 V

r ₀

^H ⁽ r - r k - 1

) -⁽¹ S r k ) r k - ⁽1 V r k J      г V

_-

k

)

_-

² и r k ) N ^r ⁽⁽ ^k - ^- ¹)'

L d m ⁽ r k - 1 )

s ( r k - 1 ) r k - ¹ ^r 0 V

_-

r k - 1

r ₀

^H ⁽ r - r k - 1

) -

N_r ( r_k )

Bending moments will have expressions

M r = d om

+ ^v 0.2

r

V

+

V

^

Mr ₀

D_OM

Qr 0 ²

4 D OM

r

_-

V

r

^

r

\

^r 0 J

^

r

r

0.43 M - 0.04 q_or _o ²

0.346

q ₀ r ₀

D_OM

^

0.236 Qr ²

\

+

^q 0 ^r 0 ³

3 D OM

■< v 2 In

D OM

^

r

r

^

+

^r 0 V

^r 0 J

0.42

Qr 0 ²

D_OM

0.43

M

^

1ln 1

^

r

\

V

r

D M ( r k ) S ( г . - 1 )

D OM

r ₀

+

V

D_OM

r

r 0 1

^

_-

r

\

V

r

^rk - 1

^r 0 J

D_M ( r_k )

0.5 Qr 0 ²

0.04

q ₀ r ₀

D_OM

^

^

^

r

V

^H ⁽ ^r ^- ^rk - 1

r 0 V

r

+ 2 - S ( r . - 1 ) r k - ¹ 1

_-

r k *1

_-

V r k J

^r 0 V

^r 0 J

^H ⁽ ^r ^- ^rk - 1

⁾ ^-

\

V r k J

² ^ ⁽ r k )

N r ⁽ r k - 1 )

^DM ⁽ ^rk - 1 ⁾

S ( r . - 1 ) ^r r^ 1

^r 0 V

_-

^rk - 1

^r 0 J

_-

N_r ( r_k )

_-

D_M ( r_k )

S ( r . ) rk- 1

r 0 V

^v J 1 ( r ) j v ² m

_-

r_k

^r 0 J

^H ⁽ r ^- r k X ^r

D M ( r . ) S ( r . - 1 )

_-

D_OM

_-

r ₀

V

S ( r k ) r 1 ^r 0 V

_-

k

) I

0.125 q r ²

^

0.166 q r ²

+

D OM

^

V

0.236

^

^

r

r

Qr ₀

D_OM

^r0 J JJ

) - In

S ( r . ) rA 1 ^r 0 V

^H ⁽ ^r ^- ^rk - 1

r k - 1 )

rk - 1

r ₀

J

_-

>

_-

D M ( r S ( r . )

D OM

r ₀

V

_-

r_k

^r 0 J

^H ⁽ ^r ^- ^rk - 1

) +

r_k

^r 0 J

^H ⁽ ^r ^- r k

)

_-

)

_-

^H(r ^- rk - 1 ^)(r ^- rk - 1 )

_-

—

ln

DM^(-k) ^ ^(r _k )

D_OM

r ₀

—

r_k

г

+ E -|X r k — 1 )

к r k

-И r k )

N r ( r k )

D M ( r k )

к

r ₀

У

^H(r ^— rk — 1

^)(r ^— rk — 1

)

+

rk — 1

N r ( Г к — i )

D M ( r k — 1 )

г

_—

rk — 1

r ₀

к

r ₀

У

^ ( - k — 1 ) ^- ^k ^— ¹- 1

r ₀

к

S ( r k ) r ^L 1

r ₀

к

_—

_—

k

H ⁽ ^r ^— ^rk ⁾⁽ ^-

) к

_—

k

r k — 1 )

_—

г

- | Х - k ) - 1 1

к r k

^r 0 к

_—

r_k

r ₀

H ⁽ ^r ^— rk ⁾⁽ ^r

_—

r k — 1 )

_—

⁾⁽ ^r ^— ^r k — 1

)

_—

У

^М Ө = D OM

у

+ ¹ 0.2

r

к

к

Mr ₀

_—

D_OM

+

Qr 0 ²

у

4 D_OM

v

к

_—

r

r

_—

r

r

<

0.43 M — 0.04 q—

0.346 ^q ⁰ ^r ⁰

к

+

_—

0.236 Qr ²

D_OM

^q 0 ^r 0 ³

3 D_OM

у

+

0.5 Qr 0 ²

_—

0.125 q r

_—

0.166 q r ²

у

•^ v Eh

^DOM

^- 1

^r 0 к

_—

_—

r

r

у

+

0.42

Qr 0 ²

D_OM

+ 0.43

M

_—

у

- Һ 1

_—

к

r

r

D M ⁽ r k ) ^ r k — 1 )

к

у

+

r

у

Г 0 1

_—

^DOM

к

_—

r

r

D_OM

r

к

_—

^- 1

_—

r

^DOM

r ₀

к

_—

^r 0 к

r

У 1 A „ У

+ E - 4r k . 1 ) r   1

_—

^rk - 1

_—

к ^rk У

^r 0 к

^r 0 У

^H ⁽ ^r ^— ^rk — 1

0.04 ^q ⁰ ^r ⁰

_—

у

D_OM

0.236

Qr ₀

D_OM

J 1 ( г ) +

-л

_—

к

r

16 1

_—

r

^rk — 1

r ₀

У

H(r

_—

к

r

J У

rk — 1

) — In

d m ⁽ r k ) - X r k )

^DOM

r ₀

у

к

_—

r_k

r

H ⁽ ^r ^— r k

) +

)

_—

^f1 I t r k ) r f 1 — r f H ( r — r k )

r

^r 0 к

r

_—

^E ^ ⁽ r k

N r ⁽ r k — 1 )

у

_—

r

d m ⁽ ^rk — 1 )

^ ( r k — 1 ) r^k — ¹ 1

^r 0 к

_—

^rk — 1

^r 0 У

^H ⁽ ^r ^— ^rk — 1

)

_—

N r ⁽ r k )

D M ⁽ r k )

f

5 ( r k ) - ^ 1

^r 0 к

_—

r_k

r

^H ⁽ ^r ^— r k

) ^

_—

s ( - k — 1)¹ ( 1

D OM

_—

^rk — 1

^r 0 к

^r 0 У

^H ⁽ ^r ^— ^rk — 1

⁾⁽ ^r ^— ^rk — 1

)

_—

- ln D M ⁽ ^rk ) X r k )— f i -— ^J H ( r - r k - 1X r - r k - i ) D OM        ^r 0 I    ^r 0 J

+

+ 2

- ^ ⁽ ^r k - 1 ) ¹ ^- — H ⁽ ^r ^- ^r k - 1X ^r ^- ^r k - 1)— ^ ⁽ ^r k )- ¹ ^- - H ⁽ ^r ^- ^r k ⁾⁽ ^r ^- ^r k - 1 )

к ^r k 7           ^r 0 V      ^r 0 7                             к ^r k 7        ^r 0 к ^r 0 7

_-

Z ^ ( r k )

N r ⁽ ^r k - 1 ) [ D M ( r k - 1 )

^ ( r k - 1 ) r k - ¹ r ₀

к

^r k - 1

^r 0 7

H ( r

- r k - 1 )( r - r k - 1 ⁾ ^-

N r ⁽ ^r k ) d m ⁽ ^r k )

^ ( r k ) r k - 1

_-

r 0 к

^r-D H ⁽ r

r ₀ 7

- r_k )

Results and their discussion

The application of the partial sampling method has made it possible to solve the problem for any law of changing mechanical characteristics. On the basis of the found solution and numerical analysis for stress-deformed state of the lithospheric plate under the action of the normal stretching effort and transverse forces symmetrically distributed in the middle plane, as well as a result of temperature heating, patterns of change of radial bending moments and circumferential bending moments are found, which are shown in the form of graphs in Figures 1.

Figure 1 - Circumferential bending moments at joint bending and stretching of non-uniform plate of variable thickness.

Conclusion

New model of stress-strain state of axisymmetric lithospheric plate of exponential profile in non-uniform temperature field and under action of normal stretching effort and transverse forces symmetrically distributed in the middle plane is proposed.

To solve the nonlinear differential equation with non-uniform bending coefficients of the lithospheric plate, the partial sampling method was applied for the first time.

Obtained regularities of change of radial bending moments and circumferential bending moments under action of normal stretching effort and transverse forces symmetrically distributed in the middle plane, and also as a result of temperature heating characterize stressed-deformed state of plate. Graphical analysis indicates the nonlinear nature of their distribution.

Список литературы Thermal deformation, complex bending and stretching of lithospheric plate in tectonic force field

Baimukhametov A.A., Koksalov K.K., Veksler Ju.A. The problems of lithospheric geodynamics//News of the National Academy of Sciences of the Republic Kazakhstan. Series of geology and technical sciences. - 2019. - Volume 5. - Number 437. - P. 91-98.
Baimukhametov A.A., Martynov N.I., Ramazanova M.A., Tanirbergenov A.G. Applied aspects of researches of mathematical modelling of salt diapirizm in oil and gas business// News of the National Academy of Sciences of the Republic of Kazakhstan. Series of geology and technical sciences. - 2017. - Volume 2. - Number 422. - P. 185 - 193.
Baimukhametov A.A., Martynov N.I., Tanirbergenov A.G. Mathematical modeling of the formation of salt domes in the Earth 's crust (monograph). Almaty, Gylym ordasy. - 2017. - 242 p..
Turekhodzhayev A.N., Mamatova G.U., Rystygulova V.B. Complex bending of elastic nonuniform plate in uneven temperature field//Journal of Karaganda University. Series Mathematics. - 2014. № 2 (74). - P.135-140..

Статья научная

Thermal deformation, complex bending and stretching of lithospheric plate in tectonic force field

Текст научной статьи Thermal deformation, complex bending and stretching of lithospheric plate in tectonic force field

П (r) = - v X ln

D N ( r k ) ■ r k - 1 )

7 ( r - r k - 1

) - In

r k - 1 ) -

^ ( r ) =

N r ( r k - 1 )

■ ( r k ) 7 ( r - r k - 1

( X t d m ); ^ ( r ) = 1 + r

■ ( r ) = 0.2

0.346

q 0 r 0

1ln 1

Jx ( r ) v X ln

ln

0.42

0.43

r o 1

16 1

0.04

q 0 r 0

0.236

J 1 ( r ) +

d m ( r k ) ■ r k - 1 )

r 0

)( r - rk - 1

r 0

r 0

r 0

rk - 1 )( r

1 ^ ( r . ) r-

r 0

>

N r ( г . - 1 )

D M ( г . - 1 )

r 0

)( r - r k - 1

DM ( rk )

■ ( r . ) - k- 1

r 0

r 0

У

к

к

+

к

+

у

+

•^ v Eh

+

к

+

к

к

У 1 A „ У

к rk У

к

У

к

J У

d m ( r k ) - X r k )

к

) +

)

f1 I t r k ) r f 1 — r f H ( r — r k )

E ^ ( r k

)

N r ( r k )

D M ( r k )

f

5 ( r k ) - ^ 1

) ^

r 0 У

)

r 0 7

r 0 к

r 0 7

П (r) = ^- v X ln

D N ⁽ r k ) ■ r k - 1 )

⁷ ⁽ ^r ^- ^r k - 1

^r k - 1 ⁾ ^-

^ ⁽ ^r ) =

^N r ⁽ r k - 1 )

■ ⁽ ^r k ⁾ ⁷ ⁽ ^r ^- ^r k - 1

( X t d m ); ^ ⁽ ^r ) = ¹ ⁺ r

q ₀ r ₀

J_x ( r ) v X ln

q ₀ r ₀

J 1 ⁽ ^r ^{) +}

d m ⁽ r k ) ■ r k - 1 )

r ₀

⁾⁽ ^r ^- ^rk - 1

r ₀

r ₀

r ₀

^rk - 1 ⁾⁽ r

¹ ^ ( r . ) r-

r ₀

r ₀

⁾⁽ ^r ^- r k - 1

D_M ( r_k )

r ₀

r ₀

к ^rk У

d m ⁽ r k ) - X r k )

^f1 I t r k ) r f 1 — r f H ( r — r k )

^E ^ ⁽ r k

N r ⁽ r k )

D M ⁽ r k )

^r 0 У

^r 0 7

r ₀ 7