О КРУЧЕНИИ ПАРАЛЛЕЛЕПИПЕДА ВОКРУГ ТРЕХ ОСЕЙ

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Introduction. Some tasks of the deformable solid body mechanics are studied rather well. These are sij - dij p = leij , (2) so-called statically definable tasks. These tasks deal with torsion of prismatic bars and with plane strain state. They belong to the wide range of tasks - limit state of deform- able bodies. The theory of the limit state is one of the fundamental sections of the deformable solid body mechanics [1]. The theory of the limit state deals with where δij - Kronecker delta; λ - unspecified nonnegative function, 3p = σij. Equation system (1)-(2) closes by Mises yield condi- tion 11 22 33 (s - p )2 + (s - p)2 + (s - p )2 + statically definable state of solid bodies. In this case the + 2 (s2 + s2 + s2 ) = 2k 2. (3) system is closed at the expense of limit conditions and such properties of matter as viscosity, elasticity, etc. can- not influence the limit state. In other words once the limit state has been achieved, the nature of relation between stress and strain does not influence the limit state. Some of such systems are considered in [1-3]. In the first part one system of plasticity equations which describes the limit state is considered. This system can be used for the description of plastic current around three orthogonal axes. Problem setting. Suppose x = x1, y = x2, z = x3 - orthogonal axes, u, v, w - components of velocity defor- mation vector, eij - components of velocity deformation 12 13 23 S It is known [1] that in case of prismatic bar torsion around oz axis, the field of deformation velocities is as folows u = -yz, v = xz, w = w(x,y). (4) Generalizing ratios (1) we will demand u = u(y,z), v = v(x,z), w = w(x,y). (5) We will construct the system of equations correspond- ing to the field of deformation velocities. As a result we receive the following system which will be researched in the study presented S tensor, σij - components of stress tensor. Components of stress tensor conforms with the equilibrium equations ¶ y t1 + ¶ z t2 = ¶ x p, ¶ x t1 + ¶ z t3 = ¶ y p, 2 2 2 (6) ¶isij = 0. (1) ¶ xt2 + ¶ y t3 = ¶ z p, (t1 ) + (t2 ) + (t3 ) = k 2. On the repeating indexes summing is supposed. De- viator of stress tensor and deformation velocity tensor are coaxial The system of equations (6) can be used, in particular, for the description of rectangular parallelogram in the plastic state torsion around three axes (see a figure.). The torsion of parallelepiped around the three axes Кручение параллелепипеда вокруг трех осей We will assume that the parallelogram twists around axes of ox, oy, oz in equal and opposite pairs of forces with the moments M1, M2, M3. At the same time there are where n1 = ¶xy , n2 = ¶ yy , n3 = ¶zy . i some limit moments M1, M2, M3 when the parallelepiped passes into the plastic state and begins to twist. From the system (6) it is visible that such task is statically definable One of solutions of the equation (11) which does not depend on values t1, t2, t3 is and can serve for limit moment values finding via formulas 2n2 = 1, i = 1, 2, 3. 1 òò M * = ( yt2 - zt1 )dydz, Therefore, an angle between the normal to a character- 2 M * = òò(zt1 - xt3 )dxdz, (7) ristic surface y ( x, y, z ) = 0 and vector n equals ± p 4 . 3 òò M * = (xt3 - yt2 )dydz. Set of elements of the characteristic surface forms the solution cone ± p 4 around the direction which is defined Except the moments (7) the body is affected by hydro- static pressure P å = P0 , ∑ - lateral surface of a parallelepiped. We will now study some properties of the system (7). 1. Characteristic surfaces of system (6). 3 The system (6) contains the finite ratio, connecting values t1 , t 2 , t . After having differentiated it on x, y, z by the third root of the equation (11) and depends on tension. Point symmetries of the equation system (6). Point symmetries are widely used in the studies of differential equations. Necessary data on symmetries and their appli- cation to the equations of plasticity and elastic plasticity can be found in [4-8]. Since the system (6) contains finite ratio, we should work with its consequences, which looks like (9), where for convenience the following designations are entered ¶ t1 = q1, ¶ t2 = q2, ¶ t3 = q3, ¶ p = q4 etc., we receive the system x 1 x 1 x 1 x 1 ¶ t1 + ¶ t2 = ¶ p, ¶ t1 + ¶ t3 = ¶ p, q1 + q2 = q4, q1 + q3 = q4, q2 + q3 = q4, ¶ x t2 + ¶ y t3 = ¶ z p, t1q1 + t2q2 + t3q3 = 0, t1¶ t1 + t2¶ t2 + t3¶ t3 = 0, (8) t1q1 + t2q2 + t3q3 = 0, (12) t1¶ y t1 + t2¶ y t2 + t3¶ y t3 = 0, t1q1 + t2q2 + t3q3 = 0, 3 3 3 y z x x z y 2 3 1 1 3 2 1 2 3 1 1 1 x x x 2 2 2 t1¶z t1 + t2¶z t2 + t3¶z t3 = 0. Let’s represent the equation of the characteristic sur- face of the equation system (9) as (t1 )2 + (t2 )2 + (t3 )2 = k 2. S We will search for point symmetries relative to which the diversity determined by the system of equations (12) y = y ( x, y, z ), (9) is invariant. According to Lie-Ovsyannikov’s technique, we will Characteristic surfaces of the system (8) are found from the determinant search for the admissible operator of point symmetry in view of ¶xy ¶ yy ¶ zy 0 ¶ yy ¶ xy 0 ¶ zy ¶zy 0 ¶ xy ¶ yy 0 t1 t2 t3 X =x j ¶ ¶xj + hi ¶ , ¶ti j = 1, 2, 3; i = 1, 2, 3, 4. (13) = 0. (10) We continue the operator (13) on the first derivatives by formulas Note. It is easy to see that all three latter equations of the system (8) give identical lines in the determinant (10). Expending the determinant (10) on the last line we where Vi = D X% = X + Vi ¶ , k k ¶qi (hi ) - qi D (zb ), D = ¶ (14) + qi ¶ . receive k k b k k ¶x k ¶ti z z x y t1¶ y ((¶ y)2 - (¶ y)2 - (¶ y)2 )+ k + t2¶ y (¶ y)2 - (¶ y)2 - (¶ y)2 + With the operator (14) we affect the system of equa- tions (13) and transfer to the diversity set by this system. As a result we receive polynoms of the second level y ( y x z ) in relation to “internal” - endogenous variables q2, q3 . x ( x y z ) “External” - exogenetic variables q , q are determined + t3¶ y (¶ y)2 - (¶ y)2 - (¶ y)2 k k = 0. 1 4 k k This equation can be written as 3 2 1 t1n3 (2n2 -1) + t2n2 (2n2 -1) + t3n1 (2n2 -1) = 0, (11) from the system (12) via endogenous variables. In the received polynoms of the second level we equate coef- ficients to zero in case of the first and second levels of endogenous variables. It allows to receive the redefined k k system of linear differential equations with respect to coefficients q2, q3 . Solving this system, obtained is the ¶ z trz = r¶r p, r¶r trq + r¶z tqz + 2trq = 0, r¶r trz + trz = ¶ z p, following result. Theorem. The system of equations (6) allows Lie t2 + t2 + t2 = k 2. rq rz qz S algebra L8, generated by operators In this case t is determined from the linear differen- X = ¶ , X = x ¶ , X = ¶ , rz i i ¶xi 4 i ¶x 5 ¶p tial equation r¶ t + r 2¶2t - t + r 2¶2t = 0. ¶ ¶ 2 ¶ 1 ¶ r rz r rz rz z rz X12 = x2 ¶x - x1 ¶x + t - t ¶t1 ¶t2 , Other functions are determined from the system (21). 1 2 3. Conservation laws of equation system (6). X = x ¶ - x ¶ + t1 ¶ - t3 ¶ , Conservation laws are applied to solutions of elastic - 13 1 ¶x 3 ¶x ¶t3 ¶t1 plasticity equations. Necessary determination and exam- 3 1 ¶ ¶ 3 ¶ 2 ¶ ples conservation laws usability can be found in [9-15]. X 23 = x3 ¶x - x2 ¶x + t - t ¶t2 ¶t3 . Let’s find conservation laws of equation system (6) 2 3 in the following view i = 1, 2, 3, x' = x exp a , shift for 1 2 3 Availability of the operators Xi, i = 1, 2, 3, 4 means that the system (6) allows shifts and stretching on axes x, ¶x A(t1, t2, t3, p)+ ¶ y B (t1, t2, t3, p)+ y, z x' = x + a , i i i i i 4 +¶zC (t , t , t , p) = 0. hydrostatical pressure p ' = p + a5 , as well as rotation around three coordinate axes. 2. Invariant solutions of equation system (6). 2.1. Let’s create invariant solution relative to subalgebra generated by the operator The equate is done on the account of equation system (6). From this follow the ratio X12 A - t2¶ p B + t1¶ pC = 0, ¶ X13B - t3¶ p A + t1¶ pC = 0, X12 B - t2¶ p A = 0, X3 = ¶z . X C - t1¶ A = 0, This type of solution should be searched in the follow- ing view 12 p X13C + t1¶ p B = 0, X13 A - t3¶ p B = 0, ti = ti ( x, y ), p = p ( x, y ). (15) where X12 = -t2¶ 1 + t1¶ 2 , X13 = -t3¶ 1 + t1¶ 3 . t t t t We add (15) in system (6) and we obtain ¶ t1 = ¶ p, ¶ t1 = ¶ p, ¶ t2 + ¶ t3 = 0, Let’s show that these equations are compatible. Sup- pose ¶ p A = ¶ p B = ¶ pC = 0, than - one of the solutions y x x y x y will be the infinite series 2 2 2 (16) S (t1 ) + (t2 ) + (t3 ) = k 2. A(S ), B (S ), C (S ), From (16) easily obtain where S = (t1)2 + (t2 )2 + (t3 )2 , A(S ), B (S ), C (S ) - t1 = f ( x + y ) + g ( x - y ), p = f ( x + y) - g ( x - y), (17) random differentiable functions. Remark. Are there other laws? Not stated, but accord- ¶ yt1 + ¶z t2 = ¶x p. Now functions t2, t3 are determines from the equation systems ing to the author other conservation laws do not exist. 4. It is clear, for the system (6) tension state is the most relevant. Supposing it is known. Than to find three components of the velocity vector we have three equa- ¶xt2 + ¶ y t3 = 0, (t2 ) + (t3 ) = k 2 - (t1) . (18) Подпись: S 2 2 2 tions t1 = le , t2 = le , t3 = le , (21) The equation system (18) describes the bar torsion in the conditions when the yield stress (limit of fluctuation) depends on variables x, y. These tasks are considered in [4] and in the literature quoted. where 12 13 23 2e12 = ¶ yu + ¶ xv, 2e13 = ¶ zu + ¶ xw, 2e23 = ¶ zv + ¶ yw. 2.2. Let’s construct the invariant decision relative to subalgebra generated by the operator X12. This operator Let’s show that the equations (21) can be solved in terms of deformation velocity tensor components. It is in cylindrical coordinate system rθz looks like X12 = ¶ . ¶q known that except the equations (21) deformation veloc- ity tensor components shall satisfy equations of compati- In this case the system (6) will be written as follows bility as well. Owing to ratios (22) and (5) only six ¶qtrq + ¶z trz = r¶r p, r¶r trq + r¶z tqz + 2trq = ¶q p, of them remain. ¶2 e = 0, ¶2 e = 0, ¶2 e = 0, r¶r trz + ¶qtqz + trz = ¶z p, (19) xy 12 xz 13 yz 23 t2 + t2 + t2 = k 2. Подпись: rq Подпись: rz Подпись: qz S ¶ x (¶ xe23 - ¶ ze12 - ¶ ye13 ) = 0, (23) Подпись: rq Подпись: rz Подпись: qz S Invariant solution in this case is determined from the following system ¶ y (¶ ye13 - ¶ ze12 - ¶ xe23 ) = 0, ¶ z (¶ ze12 - ¶ ye13 - ¶ xe23 ) = 0. Theorem. The compatibility equations of deformation speeds are done identically. In this case from (21) we have (t1)2 (e2 + e2 + e2 ) = k 2e2 , used for the description of elastic status of the parallelepi- ped twisted around three orthogonal axes. The moments are defined from formulas (7). As e11 + e22 + e33 = 0, dilatation (volume change) is equal to zero. The solution (27) 12 13 23 S 12 describes vortex movement characterized by the vector w (t2 )2 (e2 + e2 + e2 ) = k 2e2 , (24) 12 13 23 S 23 æ i j k ö 2 ç ÷ (t3 ) (e2 + e2 + e2 ) = k 2e2 . w= ç¶ x ¶ y ¶ z ÷. 12 13 23 S 13 ç ÷ Equation system (24) is a system of linear homogeneous è w1 w2 w3 ø 12 13 23 equations relevant to variables e2 , e2 , e2 . Its determinant is 2 2 2 Herewith movement paths will be vortex lines which are defined from the equation S (t1 ) - k 2 (t1 ) (t1 ) dx = dy = dz . S (t2 )2 (t2 )2 - k 2 (t2 )2 . w1 w2 w3 S (t3 )2 (t3 )2 (t3 )2 - k 2 Conclusion. In the present work for the first time con- sidered is the system which can be used for the analysis of stress state appearing under torsion of the parallelepiped e 23 This determinant equals zero as the amount of all lines is equal to zero. It means that the system (24) has only two independent equations for three components of a around three orthogonal axes. At that it can be in either plastic or elastic state. deformation speed tensor. For example, value 2 can References be picked up randomly, thus for the given tension state, defined from the system (6), velocity field is defined with the functional arbitrariness. 5. In this part we will consider three-dimensional equations of elasticity in static. The system of equilibrium equations is described using equations (1), relation be- tween components of stress tensor and deformation tensor is as follows e = (s11 - v (s22 + s33 )) , 1. Ivlev D. D., Maksimova L. A., Nepershin R. I. et al. Predel’noe sostojanie deformirovannyh tel i gonnyh porod [The ultimate state of deformed bodies and rocks]. Moscow, Fizmatlit Publ., 2008, 832 p. 2. Ivlev D. D. Teoriya ideal’noy plastichnosti [Theory of ideal plasticity]. Moscow, Nauka Publ., 1966, 232 p. 3. Ishlinskij A. Ju., Ivlev D. D. Matematicheskaya teoriya plastichnosti [Mathematical theory of plasticity]. Moscow, Fizmatlit Publ., 2001,701 p. 11 e22 = E (s22 - v (s11 + s33 )) , E (25) 4. Ol’shak V., Ryhlevskij Ja., Urbanovskij V. Teoriya plastichnosti neodnorodnyh tel [The theory of plasticity of inhomogeneous bodies]. Moscow, Mir Publ., 1964, 156 p. e = (s33 - v (s22 + s11 )) , 5. Ovsyannikov L. V. Group Analysis of Differential 33 e = s12 , 12 2m E e = s13 , 13 2m e = s23 , 23 2m Equations. Academic Press, NewYork, 1982. 6. Senashov S. I., Yakchno A. N. Reproduction of solutions for bidimensional ideal plasticity. Journal of Non-Linear Mechanics. 2007, Vol. 42, P. 500-503. where eij deformation tensor components, E,m, v elastic constants. Supposing vector deformation components are as follows 7. Senashov S. I., Yakchno A. N. Deformation of characteristic curves of the plane ideal plasticity equations by point symmetries. Nonlinear analysis. 2009, Vol. 71, P. 1274-1284. 8. Senashov S. I., Cherepanova O. N. [New classes w1 = w1 ( y, z ), w2 = w2 ( x, z ), w3 = w3 ( x, z ). Inserting (26) into (25) we obtain (26) of solutions of the minimal surface equations]. Journal of Siberian Fed. Univ., Math. & Ph. 2010, No. 3(2), P. 248- 255 (In Russ.). ¶ w + ¶ w = s12 , ¶ w + ¶ w = s13 , 9. Senashov S. I Conservation Laws, Hodograph y 1 x 2 2m z 1 x 3 2m s (27) Transformation and Boundary Value Problems of Plane Plasticity. SIGMA. 2012, Vol. 8, P. 16. m ¶zw2 + ¶ yw3 = 23 . 2 In this case equation (1) with regard to (26), (27) is as follows 10. Senashov S. I., Yakchno A. N. Some symmetry group aspects of a perfect plane plasticity system, J. Phys. A: Math. Theor. 2013, Vol. 46, No. 355202. 11. Senashov S. I., Yakchno A. N. Conservation Laws ¶ yy w1 + ¶ zz w1 = 0, ¶ xx w2 + ¶ zzw2 = 0, of Three-Dimensional Perfect Plasticity Equations under ¶ xx w2 + ¶ yy w3 = 0. (28) von Mises Yield Criterion. Abstract and Applied Analysis. 2013, Article ID 702132, 8 p. It is shown that components of deformation vector are harmonic functions. The solutions obtained here can be 12. Senashov S. I, Kondrin A. V., Cherepanova O. N. On Elastoplastic Torsion of a Rod with Multiply Connected Cross-Section. J. Siberian Federal Univ., Math. & Physics. 2015, No. 7(1), P. 343-351. 13. Senashov S. I., Cherepanova O. N., Kondrin A. V. Elastoplastic Bending of Beam. J. Siberian Federal Univ., Math. & Physics. 2014, Vol. 7(2), P. 203-208. 14. Senashov S. I., Filyushina E. V., Gomonova O. V. Sonstruction of elasto-plastic boundaries using conservation laws. Vestnik SibGAU. 2015, Vol. 16, No. 2, P. 343-360 (In Russ.). 15. Senashov S. I., Vinogradov A. M. Symmetries and Conservation Laws of 2-Dimensional Ideal Plasticity. Proc. of Edinb. Math. Soc. 1988, Vol. 31, P. 415-439.

Об авторах

С. И. Сенашов

Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева

Email: sen@sibsau.ru
Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31

И. Л. Савостьянова

Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева

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

Е. В. Филюшина

Сибирский государственный университет науки и технологий имени академика М. Ф. Решетнева

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

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