Cauchy problem solution for a hyperbolic system of the homogeneous 2-dimensional quasilinear equations Текст научной статьи по специальности «Математика»

UDK 539.374

S. I. Senashov, A. Yakhno

CAUCHY PROBLEM SOLUTION FOR A HYPERBOLIC SYSTEM OF THE HOMOGENEOUS 2-DIMENSIONAL QUASILINEAR EQUATIONS*

The method ofsolving the boundary-value problems for hyperbolic system of the homogeneous quasilinear equations of two independent variables with the help of conservation laws is presented. This method is applied to basic boundary problems for the system of two-dimensional plasticity equations under Sent-Venan—Mises yield criterion, as well for the system under Coulomb’s criterion.

Keywords: conservation laws, plane ideal plasticity.

Among hyperbolic systems of the nonlinear equations with partial derivatives the systems of the quasilinear equations of two independent variables, are most studied. These systems in particular describe the unsteady onedimensional and the supersonic two-dimensional stationary flows of compressible gases and liquids, two-dimensional deformed plastic state of continuous medium, etc. A lot of them are reduced to a hyperbolic system of homogeneous quasilinear equations

ux + A(u, v)uy = 0,

vx + B(u, v)vy = 0, (1)

where u = u( x, y), v = v( x, y), an indices the lower meaning of the derivation with respect to the corresponding variables. This kind of system can be linearized by applying a so-called hodograph transformation if and only if the corresponding Jacobian is not equals to zero in some domain of solution existence.

From the other hand, the conservation laws [1] are one of the fundamental characteristics of any mechanical process. In the section 2 we have described a method of analytical solving of boundary-value problems for the system (1). This method is based on the use of conservations laws of systems under consideration. The main features of the presnted method consist of the possibility of problem linearization without considering the singularity of Jacobian transformation to obtain exact solutions for boundary-valued problems in explicit form.

In section 3 we consider the application of this method to some exact systems of the bi-dimensional plasticity mathematical theory.

Conservation Laws. Let us set Cauchy’s problem for the system (1): in something like an arc a <t<b of a smooth curve L

L = {(x,y): x = x(t),y = y(t),t e [a,b]} in the plane xOy it is required to find a solution of the system (1), that takes given values on L

u(x,y) Il = u(x(t),y(t)) = u0(t), v(x, y) Il = v(x(t), y(t)) = v0 (t).

The characteristics equations of the system (1) look like

dx = A. d-x = B, (2)

dy dy

with the relations in the characteristics u = u0, v = v equally.

A conservation law of the equations set (1) is searched in the form of the relation

C + Dy = 0, (3)

where C = C(u,v), D = D(u, v) should vanish from all solutions of the system (1):

Cuux + Cvvx + Duuy + Dvvy =

= -ACuuy - BCvvy + Duuy + Dvvy = 0

hence

Du - ACu =0, Dv - BCv =0. (4)

Equation (3), if the conditions of Green’s theorem are satisfied, is equivalent to a relation

I - Cdy + Ddx = 0,

where r is an arbitrary smooth closed contour.

In the plane xOy we have considered the closed path MNK, where M(xm (a), ym (a)), N(xn (b), yn(b)) e L, K ( xk, yk ) is the point of an intersection of characteristics v = v0, u = u0, drawing through the points M, N respectively.

Then,

| - Cdy + Ddx = | Ddx - Cdy +

MNK MN

+1 Ddx - Cdy + | Ddx - Cdy = 0.

(5)

NK KM

Taking into account expressions (2)

| Ddx - Cdy = | (D - AC)dx =

NK NK

■ x(D - AC)|x* - | xd (D - AC)dv.

n J

Similarly,

| Ddx - Cdy = | (D - BC)dx =

KM KM

= x(D - BC) О" - I xdu (D - BC)du.

KM

Let us put the following conditions

d v = 0,5,

( D - AC )| o...w.,

u =u (x(o),y(b))

(D - BC) | 0(()())

v=v (x (a), y (a))

= 0.

(6)

‘Работа выполнена в рамках Федаральной целевой программы «Научные и научно-педагогические кадры инновационной России» на 2009-2013 гг. (№ 1121) и «Развитие научного потенциала высшей школы» № 2.1.1 (3023).

NK

Let us denote

f(u, v) = D - AC, y(u, v) = D - BC.

Then

D = (Ay- Bf)/(A - B), C = (y- f )/(A - B), where A ^ B, because the considered system (1) is a hyperbolic one and has two different characteristics.

In the new variables, the system (4) has the form

fu + K1(y-f) = 0,

y v + K 2(y-f) = 0, (7)

where Kx = Au/(A - B); K2 = Bv/(A - B).

Conditions (6) we shall take as follows

f

| = 0 = Constj =1, y | = 0 = Const2 = 0.

(8)

Coming back to (5) and taking into account (8), we shall obtain

f Ddx - Cdy = -(x(D - AC) P + x(D - BC) |x“) =

J n k

= -( xkf |

-xnf |

+xm У |

~xt У |

>) =

= y

A

yk

Similarly,

| Ddx - Cdy = | ( — - CI dy =

KM KMè B 0

"i y5u I B

У I du.

Let’s assume

u=u0 ( x (b), y (b ))

5,

= 0,

= 0.

v0( x ( a), y ( a ))_

The conditions (10) we can note are the following

f I o = A(u0,v), y | o = 0.

u=u v=v

From (5) taking into account (10) we have

| Ddx - Cdy = -yk a

- y

y

+ ykB

v0 B

f

+ y"A

'■yn - yk.

Examples of applying the method. In this work [3] we have found all the conservations laws of the bi-dimensional plasticity system under the Sent-Venan-Mises yield criterion:

sx - 2k (9x cos29 + 9y sin29) = 0,

cty - 2k(9x sin29-9y cos29) = 0, (13)

where s is the hydrostatic pressure; 9 is the angle between the first main direction of a stress tensor and the ox-axis; k is a constant of plasticity. Here we have described the special class of conservation laws that are used for solution for Cauchy problem system (13).

The system (13) in the form (1) looks like this:

Xx +Xy tg 9 = 0, hx -hy ctg 9 = 0,

where 29 = h-X; CT = k(h + X). The solution of the problem (7), (8) has a form f = p/cos 9, y = 2pX/ sin 9 where:

PCX, h) = R(X x0> h h0)cos

h0 -X0

(9)

If we find a solution of the linear system (7), that satisfied the boundary conditions (8), then we shall define the coordinate xk from the equation (9).

On the other hand, for y-coordinate

J Ddx - Cdy = J[ D - C ] dy =

NK NKè A 0

J y5v I A

\TV è 0

- 21 R(X x0> ht)sin ]d x

h0 è 0 Accordingly, the solution of the problem (7), (11) is:

P(X, h) = R(X, X0, h, h0)sin 'h0-X°

1

+—

2

IR(X,X>,ht)cos( ]dt

(10)

(11)

(12)

The solution of the problem (7), (11) makes possible to find the coordinate yk from the equation (12). Thus, we shall define the coordinates of the point K, where the values of the functions u, v can be restored.

Let us notice, that the same steps can be applied for the Riemann problem [2].

h0

where Rh,^h0) = I0 (si(X-X0)(h-h0)) is the Bessel function of the zero order of an imaginary argument;

10(0) = 1,10' (0) = 0.

The generalization of (13) shows that it is a system of ideal plasticity equations under Coulomb’s yield criterion of the form:

ctx (1 + cos 2a cos 2©) + ct cos 2a sin 2© =

= 2(ct cos 2a+ k sin 2a)(©x sin 2© -©y cos 2©), sx cos 2a sin 2© + s y (1 - cos2a cos2©) =

= -2(s cos2a + k sin 2a)(©x cos2© + ©y sin 2©),

where p/2 - 2a is a constant angle of internal friction; © = 9+p /4. Ifa = p /4. Then we have the system (13). In the system described above there is a stress state of granular materials.

This system in the form (1) looks like this:

Xx + Xytg (©-a) = 0, hx - hytg (© + a) = 0,

where X = 2 tg2a ln(s ctg 2a + k) -©;

h = 1tg 2a ln(s ctg 2a + k) + ©. The solution of problem (7), (8) has a form [4]:

f = Y (-X, -h)V (X, h)/ cos(©-a),

g(X, h) = exp(-(X + h)/2ctg 2a),

where

V = Y(Xo,h0)R(X,Xo,h,h0)cos((h0 -X0)/2 -a)-

-2 fR(X, Xo, h, t)YC^ t) x

h0

x [sin ((t -X0)/2 - a) - ctg2a cos ((t -X0) / 2 - a)] d t. (14) The solution of the problem (7), (11) is:

V = h0)R(X Xo, h, h0)sin ((h0 -X0)/2 -a)- 2 f R(X, Xo, h, t)g (Xo, t) x h0

x[ctg 2a sin ((t — X0) / 2 -a) - cos ((t — X0) / 2 - a)] d t, (15)

where R(X,h,Xo,ho) = Io (V(X-Xo)(h-ho)/sin2a). From the equation (9) taken into account (14) we have discovered coordinate xk. Using (15), and from equations (12) we have received coordinate yk. This way, we have determined point K, in which the values of functions X, h are restored.

The bibliographic list

1. Symmetries and conservation laws for differential equations of mathematical physics : Monograph / A. V Bocharov,

V N. Chetverikov, S. V. Duzhin, et al. Amer. Math. Soc., 1999.

2. Senashov, S. I. The solving of the main boundary problems of plasticity by means of conservation laws. Modern Group Analysis VII, Developments in Theory, Computation and Application / S. I. Senashov, A. N. Yakhno. Norway : MARS Publishers, 1999. P. 149-154.

3. Senashov, S. I. Simmetries and conservation laws of 2dimensional ideals plasticity / S. I. Senashov, A. M. Vinogradov // Proc. Edinburg Math. Soc. 1988. Vol. 31. P. 415-439.

4. Kiriakov, P. P. Applications of symmeties and conservation laws for solution of differential equations / P. P. Kiriakov, S. I. Senashov, A. N. Yakhno // Publ. of Siberian Branch ofRussian Academy of Science. 2001. (In Russian).

С. И. Сенатов, А. Яхно

РЕШЕНИЕ ЗАДАЧИ КОШИ ДЛЯ ГИПЕРБОЛИЧЕСКОЙ СИСТЕМЫ ОДНОРОДНЫХ ДВУМЕРНЫХ КВАЗИЛИНЕЙНЫХ УРАВНЕНИЙ

Описан метод решения граничных задач для гиперболической системы однородных квазилинейных уравнений двух независимых переменных с применением законов сохранения. Этот метод применяется к задаче Коши для системы двумерной пластичности с условием Сен-Венана-Мизеса, а также для системы с условием Колумба.

Ключевые слова: законы сохранения, двумерная идеальная пластичность

© 8гпа.Жоу 8.1., Уаккпо А., 2009

УДК 539.3

А. В. Лопатин, Р В. Авакумов

УСТОЙЧИВОСТЬ ОРТОТРОПНОЙ ПЛАСТИНЫ С ДВУМЯ СВОБОДНЫМИ КРАЯМИ, НАГРУЖЕННОЙ ИЗГИБАЮЩИМ МОМЕНТОМ В ПЛОСКОСТИ*

Решена задача устойчивости при чистом изгибе ортотропной пластины, у которой два противоположных края свободны, а два других края шарнирно закреплены. Для решения задачи использовался метод конечных разностей.

Ключевые слова: ортотропная пластина, метод конечных разностей.

Задача устойчивости прямоугольной пластины, нагруженной по двум противоположным краям усилиями, распределенными по линейному закону, впервые была решена для изотропной пластины И. Г Бубновым [1] и С. П. Тимошенко [2]. Для ортотропной пластины эта задача была впервые сформулирована и решена С. Г Лех-ницким [3]. Эти классические решения были получены для случая шарнирного закрепления краев пластины в форме двойных тригонометрических рядов. Для определения критического усилия в этих решениях был использован энергетический метод Ритца. Это связано с тем,

что дифференциальное уравнение устойчивости пластины имеет переменный коэффициент и поэтому его интегрирование оказывается затруднительным. Метод Ритца был также использован в [4] и [5] для решения рассматриваемой нами задачи применительно к композитным пластинам с шарнирно закрепленными краями, которые нагружены равномерными сжимающими усилиями.

Таким образом, можно утверждать, что устойчивость пластины при изгибе в плоскости наиболее полно исследована только для самого распространенного вида граничных условий - шарнирного закрепления сторон. Это

‘Работа выполнена в рамках Федеральной целевой программы «Научные и научно-педагогические кадры инновационной России» (НИР НК-86П/7).