New exact solutions which describe 2-dimensional velocity field for prandtl's solution Текст научной статьи по специальности «Математика»

O. V. Gomonova, S. I. Senashov Siberian State Aerospace University named after academician M. F. Reshetnev, Russia, Krasnoyarsk

NEW EXACT SOLUTIONS WHICH DESCRIBE 2-DIMENSIONAL VELOCITY FIELD FOR PRANDTL’S SOLUTION*

New velocity fields are found for the well-known Prandtl’s solution which describes pressing of a thin layer ofplastic material between two parallel stiff and rough plates. The method of construction of other velocity fields is considered.

Keywords: ideal plasticity, velocity field, Prandtl’s solution.

The 2-dimensional ideal plasticity equations in case of steady-state problem have the form:

R/ -.7 T Rf ™ Rf • ™ i ^

-----2kI — cos29+--------sm29 | = 0,

Rx XRx Ry

R/ -.7 I Rf • ™ R9 r*r\ i ^ --------2k I — sin 29--------------cos 29 | = 0,

Ry XRx Ry

(1)

Rv,+ °i U tg20+fc

Ry Rx J X Rx Ry

= 0,

Rv„

= 0,

(2)

/ y = - p ~ kx, 4=ky,

(3)

y

—^IRK + ^ Rx Ry J 1 Ry Rx

Rvy

= 0.

(4)

Rvx

Rx Ry

One can see that because of its linearity the system (4) has an infinite set of solutions which can be used for the analysis of the stress-strained state of a plastic medium.

At the present moment two classes of solutions of this system are known, Nadai’s solution [1] and Ivlev-Senashov’s solution [2],which are the following:

vx = -axy + Mx - a arcsin y -

-aW 1- y2 - 2PV1 - y2 +

vy =aX T+“yr VMy+C2,

here a, M, C1, C2 are arbitrary constants (if a = 0 Nadai’s solution comes out).

Let’s point out others solutions of the system (4). Notice that in variables &, " , where / = k (& + "), 29 = &-", the equations (2) are written as follows (5):

Rv Rvy Rvy Ry

—- - tg 9—- = 0, —- + ctg 9—- = 0.

R& R& R" R"

If we put new variables into (5) using the formulas:

vx = u cos 9- v sin 9,

Rvx

Rx Ry

here / is hydrostatic pressure, 9 is the angle between the first principal direction of stress tensor and axis Ox, k is plasticity constant, vxvy are components of velocity vector ofstrainfield.

Prandtl’s solution is one of the practically applied and frequently used in different computations. This solution describes in particular the pressing of a thin layer of plastic material between two parallel stiff and rough plates, and it has the following form:

/ x = - p - k (x - 2^1 - y2

v = u sin 9+ v cos t

we ’ ll get a system (7):

Rv 1 0 Ru 1 0

-------------u = 0,----------------------v = 0.

R& 2 R" 2

(5)

(6)

(7)

p is an arbitrary constant.

It is well known that to describe the plasticity state of material completely one should know velocity field.

Let’s substitute the equations (2) into the system(1). We

get:

Further we use the following procedure: we solve the system (7), put expressions from Prandtl’s solution for & and " into this system, make substitution (6) and find a velocity field which corresponds to the solution (3).

Let’s do it, for example, using the simplest solution of the system (7). It is obvious that

v = u = exp-2 (& +") is a solution of the equations (7). Put it into the (6). We get:

vx = (cos 9- sin 9) exp 2 (&+"),

vy = (cos 9 + sin 9) exp-2- (&+").

From Prantl’s solution (3) one can get easily

&+" = -p-x-sin29, cos29 = y. (8)

k

And finally we find a new velocity field:

vx = exp 111-p - x - sin29^|^|(cos 9-sin 9),

vy = exp T 21-p - x - sin29))(cos 9 + sin 9).

By usingthis scheme some othervelocity fields are found.

For the equations (7) solutions are given in[3]. Further with respect to these indicated solutions, 5 more classes of new solutions of the equations (5) are built.

1) u = cos I Z

& + "

A cos I —"| + B sin j —"

* The work was carried out with support from FOP “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” for 2009-2013 years.

Here A, B, [, Z are arbitrary constants; [2 -22 = 1. In this case

Ru Z . T &+"

— = — sin I Z--------

c" 2 X 2

A cos [[&2" j + B sin

From the equations (7) we get

v = -Z sin I Z

+[cos I Z

& + 2

&+"

A sin I ^ "')-B cos I [& 2 "

&-"

A sin I " |-B cos I [

2

&-"

Making substitutionfor u, v into the equations (6) and

& - "

taking into account the equalities (8) and 9 = —^~ we get

vx = cos (2£--x- sin29)) [ A cos ([9) - B sin ([9)] cos 9 -- (-Z sin (-2 IL-x- sin 29^j^|: A cos ([9) - B sin ([9)] +

+[ cos (2 (- IL-x- sin 29^ [-A sin ([9) - B cos ([9)] j sin 9, vy = cos (2 ^-k - x-sm29jj[A cos ([9)-) sin ([9)] sin 9 +

+ -Z sin (2 (-p-x- sin29jj[ A cos ([9)-B sin ([9)] + + [cos (2 ^-E- - x - sin 29jj >-A sin ([9) B cos ([9)]J cos 9. 2) u = sinfz&±"

A cos I [& _ ") + B sin T& "

2 J X 2 Here A, B, [, Z are arbitrary constants; [2 - Z2 = 1. In this case

v = -Z cos I Z

&+i

+ [sin I Z And

& + i

A cos I [& 2 " j + B sin jV & 2 " A sin I [2-"^-B cos

vx = sin (2 ^-k - x -sin 29 jj [ A cos ([9)- B sin ([9)] cos 9-

-Zcos(2^-k.-x-sin29 | |[Acos([9)-Bsin([9)] +

+ [sin(2(-kp-x-sin29 \ |[-Asin([9)-)cos([9)] |sin9, vy = sin (2 ^--x _sin29 \ |[ A cos ([9)-) sin ([9)] sin 9 +

+(zcos(Zp-x-sin29 \ |[Acos([9)-)sin([9)] +

+ [sin(2^-p-x-sin29 | |[-Asin([9)-)cos([9)] |cos9.

3) u = exp I Z

& + "

A cos I [~"] + B sin f&y"

Here A, B, [, Z are arbitrary constants; [2 +Z2 = 1. In this case

v = Z exp ^2 + [expI Z

&+"

2

&+"

A cos I [ & 2 " | + B sin ^[ & 2 " A sin I [&—"j-B cos {[-—"

+[ exp

<[A cos([9)-B sin([9)] cos9-

-Z exp(H-p - x - sin29

X2X k

[A cos([9)- B sin([9)]-

jA

2k 2 2

[-Asin ([9)- B cos ([9))

sin 9,

2( p

vy = exp I —I - x - sin29

y X2X k

x[ A cos ([9) - B sin ([9)] sin 9 +

+ ( 2exp(2f -p-x-sin29

X X2X k

x[ A cos ([9)- B sin ([9)]-+[exp(-2X-p-x-sin29'| |[-Asin([9)-Bcos([9)] |cos9.

4) u = exp I 2

&+"

A exp I [& 2 "J + B exp

Here A, B, [, 2 are arbitrary constants; 22 -[2 = 1. In this case

v = 2 exp I Z

&+"

A exp I [&—"] + B exp I [&—"

+ [exp I 2

& + "

- A exp (X[&72" j + B exp v = exp [|X-p - x - sin 29^|^jx

x[ A exp (-[9) + B exp (-[9)] cos 9-

-X-Z exp X-^X-k - x - si„29JJx

x[ A exp (-[9) + B exp (-[9)] +

+[exp| ^2(^--E.-x-si„29JJ[-Aexp(-[9) + Bexp(-[9)] |

(2 ( p . vy = exp^I - x -sin29

. X 2 X k

x[ A exp (-[9) + B exp (-[9)^ sin 9 -

x[A exp (--9) + B exp (--9)] +

+— exp p - x - sin 29^j*| [-A exp (--9) + B exp (-—9)) ) cos 9.

5) u = exp | #

2+3

A1-3 + b

Here A, B, # arearbitraryconstants; #2 = 1. In this case

v = # exp| #

2+3

- A exp | #

2+3

A1-3+b

= exp" p - x - sin 29 11[-A9 + B]cos 9-

#exp| p-x-sin29 | |[-A9+ B]-

-A exp" 21+ - k- - x - sin29

sin 9,

= exp| #"--p-x-sin29 | |[-A9 + B]sin9 + +| #exp| ~f-p-x-sin29 | |[-A9 + B]-

-Aexp| #|-p-x-sin29

Bibliography

cos 9.

1. Sockolovsky, V V Plasticity Theory/W Sockolovsky. M.: HigherSchool, 1969. (inRussian)

2. Limiting State of Strained Media and Rocks / D. D. Ivlev, L. A. Maximova, R. I. Nepershin et al. M. : Physmathlit, 2008. (inRussian)

3. Polyanin, A. D. Reference Bookon Nonlinear Equations ofMathematicalPhysics /A.D. Polyanin. M.: Physmathlit, 2001. (inRussian)

© Gomonova O. V., Senashov S. I., 2009

P. V Galushin, E. S. Semenkin Siberian State Aerospace University named after academician M. F. Reshetnev, Russia, Krasnoyarsk

THE ASYMPTOTIC PROBABILISTIC GENETIC ALGORITHM*

This paper proposes the modification of probabilistic genetic algorithm, which uses genetic operators, not affecting the particular solutions, but the probabilities distribution of solution vector’s components. This paper also compares the reliability and efficiency of the base algorithm and proposed modification using the set of test optimization problems and bank loan portfolio problem.

Keywords: probabilistic genetic algorithm, mutation, selection.

The probabilistic genetic algorithm (PGA) is an attempt to create an algorithm with a scheme similar to that of the traditional genetic algorithm (GA), preserving the basic properties of the genetic operators, but defined in terms of the pseudo-Booleanoptimizationtheory [1].

The probabilistic genetic algorithm explicitly (as opposed to the traditional GA) computes the components of the probability vector and has no crossover operator (it is replaced a by random solution generation operator) but retains the genetic operators of mutation and selection.

The purpose of this study is to develop a probabilistic genetic algorithm modificationwith mutation and selection operators, effecting not particular individuals, but genes’ values distribution as a whole; and to compare efficiency and reliability ofbasic algorithm and modification.

Asymptotic mutation. PGA uses a standard GA mutation operator, which inverts genes with a given probability (as a rule, this probability is very low). Since genes mutate

independently, we can study one particular gene. All following formulas will stand for every gene in the chromosome. Let us suppose that p - denoting the probability of that fact was equal to 1 before mutation. We will determine the probability as equal to 1 for same gene after mutation (p denotes this probability). The mutation probability is pm.

The gene canbe equalto 1 aftermutationintwo cases: it was equal to 1 before mutation and has not mutated or it was equal to 0 before mutation and has mutated. If x denotes the gene value before mutation and y - after mutation - the following equality is:

p {y=; y=1; - pm )+p {x=0}pm =

= p(1 - pm )+(1 - p)pm = pm +p(1 - 2pm ).

Using the aforementioned designations for genes probabilities before and after mutation we can write down:

p' = pm +p(1 - 2pm )-

* This work was financially supported by the State programs “The Development of Scientific Potential of Higher Education Institutions” (Project2.1.1/2710) and “The Scientific andEducational StaffofInnovative Russia” (ProjectNK-136P/3).