Transformations of the generalized Hooke''s law composed on the main areas into Hooke''s law for the areas of arbitrary directions Текст научной статьи по специальности «Математика»

Актуальные проблемы авиации и космонавтики - 2016. Том 1

УДК 539.3

ПРЕОБРАЗОВАНИЯ ОБОБЩЕННОГО ЗАКОНА ГУКА СОСТАВЛЕННОГО НА ГЛАВНЫХ ПЛОЩАДКАХ В ЗАКОН ГУКА ДЛЯ ПЛОЩАДОК ПРОИЗВОЛЬНОГО НАПРАВЛЕНИЯ

Р. О. Яковлев, Д. И. Быстров, Р. А. Сабиров Консультант по иностранному языку - И. Н. Шостак

Сибирский государственный аэрокосмический университет имени академика М. Ф. Решетнева

Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31

E-mail: [email protected]

Рассматривается преобразование закона Гука для изотропного материала, записанного для главных площадок в шесть уравнений для произвольной системы координат. Уравнения находятся без геометрических схем с помощью преобразования тензора при повороте системы координат. Данный метод не применяется в основной литературе, хотя является, на наш взгляд, более удобным и простым.

Ключевые слова: тензорные преобразования, закон Гука, сопротивление материалов.

TRANSFORMATIONS OF THE GENERALIZED HOOKE'S LAW COMPOSED ON THE MAIN AREAS INTO HOOKE'S LAW FOR THE AREAS OF ARBITRARY DIRECTIONS

R. O. Yakovlev, D. I. Bystrov, R. A. Sabirov Foreign Language Supervisor - I. N. Shostak

Reshetnev Siberian State Aerospace University 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation E-mail: [email protected]

The transformation of Hooke's low for isotropic material recorded for the main areas into six equations for an arbitrary coordinate system is considered in this paper. The equations are without geometrical scheme due to the tensor's transformation by turning the coordinate system. This method is not used in basic literature though in our opinion is more convenient and simple.

Keywords: tensor transformation, Hooke's law, the resistance of materials.

In the books on strength of materials [1; 2] for an isotropic material there are three equations of the generalized Hooke's law for tension and compression on the main platforms. Then examines the state of pure shear, expressing the relation between tangential stress and relative shear equality

Y = T / G, (1)

Here G is constant. This constant is determined by geometric transformation of the element that has a square shape. Calculating extension and compression of distorted element diagonals and considering balance of element in pure shear and Hooke's law ratio at extension, determine a constant of isotropic material is determined

G = E / [2(1 + ц)]. (2)

Where G is the shear module of the material. Methods of strength of materials must provide strength, rigidity and stability of structures and details. Submission forms of education material are based on physical experiments and deformations models, which must be reliable and depend on standard and practice.

Секция «Механика конструкций ракетно-космической техники»

Transform three equations of Hooke's law recorded on the main areas in system of six equations for arbitrary coordinate system. In book [3] we find tensor's transformation method by turning coordinate system

(3)

Where and tensors are object (deformation and stress) in the original and turned coordinate system. Tensor is an object that connects original and turned coordinate system. When axis make a move around axis z on a angle coefficient get value

aii = a22 = cos a = c, a12 = sin a = s , a21 = - sin a = -s ,

T' T' 12 T' 1 a11 a12 a13 "Tn T12 T13 " a11 a21 a31

T' T 21 T' 22 T' = a21 a22 a23 T21 T22 T23 a12 a22 a32

T' J31 T' 32 T' ^33 _ a31 a32 a33 _ /31 T32 T33 _ _a13 a23 a33 _

' 12

' 21

a33 = 1 , °13 = °23 = °31 = °32 = 0 .

Transform main deformations s1; s2, s3 oriented in axis O123 to components of the deformation tensor placed in Oxyz axis:

(4)

S S S

x xy xz

S „, S „ S „,

Xy y yz

S S S

xz yz z

c 5 0" "S1 0 0" c - 5 0" 2 2 S1c +S2 5 (S2 -S1 )5C 0

= - 5 c 0 0 S2 0 5 c 0 = (S2 -S1 )5C 2 2 S15 +S 2C 0

0 0 1 0 0 S3 _ 0 0 1 0 0 1

Place 3 equation of Hooke's written in the main axis system O123:

si = [Ci - M(c2 + CT3)] /E, S2 = [C2 - M(c3 + CTi)] /E, s3 = [C3 - M(ci + ct2)] /E,

Where: E- Young's module, Poisson's ratio. Hence, in (4) we have: sX = [CTiC2 + ^2s2 - M^iS2 + CT2C2) - IMC3 ] / E, sy = [C2C2 + CiS2 - ^(C2s2 + ^c2) - ^ ] / E ,

sxy = (i + M)(C2 -Ci)SC / E , sz =s3 =[C3 -M(ci +C2)] / E , sxz = ^ syz =

Turn areas where main stress interacts on alpha angle.

ciC2 +c2s2 (c2 -oi )sc ° (c2 -ci )sc cis2 +c2c2 0

(5)

(6)

x T xy T xz C 5 0" 0 0 " C - 5 0"

Txy ст у T yz = -5 C 0 0 0 s C 0 =

T xz T yz z _ 0 0 1 0 0 ct3 _ 0 0 1

Where

CTx = CT1c2 + CT252 > CTу = CT2^ + CT152 , CTz = CT3,

(7)

Txy =(C2 -Ci)sC , Txz = ^ Tyz = °.

Place right parts in (7) to (6) and get six equations in arbitrary coordinate system turned on angle:

:[Cx -M(cy +C z )] / E , sy =[Cy -M(cz +Cx)] / E , s Z =[CZ -M(cx +Cy )] / E , (8)

s xy = (i + M) t xy / E , s xz = °, s yZ = 0. (9)

S =

0

0

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Dependency and the equation in 1'st ratio of (9) gives formula, where G is constant and equals. Turning system of relatively to axis x gives. Turning over y gives. Thus, dependences (8) and (9) can be found without geometric scheme applied in the books on strength of materials.

References

1. Timoshenko S. P. Mechanics of materials. Vol. 1. Elementary theory and problems. M.-L. : Ogiz, GOSTEKHIZDAT, 1945. 320 p.

2. Pisarenko G. S., Yakovlev A. P., Matveev V. V. Handbook on strength of materials. Kiev : Nauk. Dumka, 1988. 736 p.

3. Maze J. Theory and problems of mechanics of continuous media. M. : Mir. 1974. 319 p.

© Bystrov D. I., Yakovlev R. O., Sabirov R. A., 2016