Unloading wave in the cylindrical network from nonlinear elastic fibers

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Introduction1 The equation of movement of [1] networks in space has a form, constructed on the basis of the theory of Rahmatullin. In articles [2-7] waves in networks in rectangular Cartesian system of coordinates were investigated. Here waves in a cylindrical system of co-ordinates are investigated. Obviously, during stretching a cylindrical network is going to be narrowed. Being placed on a rigid pipe during motion, it will be exposed to operate a force of a friction in cylindrical networks, it is attempted to solve a problem about continuous waves. 1. The general equations of movement of a network The equation of motion of the network, taking into account the reaction of the supporting body and the geometric relations will have the form, in contrast to [2]. r between it and a pipe. In order to avoid it, the net- ¶ r ¶ r ¶2r r work is replaced on a screw pipe of a special profile. Such pipes are applied, in particular at the process of ¶s1 (σ1τ1 ) + ¶s2 (σ2 τ2 ) = (ρ1 +ρ2 ) 2 § pn; ¶t drilling of chinks. In practice, these phenomena can r r take place in the flexible pipelines. (1+ e ) τ = ¶r ; (1+ e ) τ = ¶r . (1) r r Aim is research of waves in cylindrical sets. 1 1 ¶s 2 2 ¶s 1 2 Considering sets of variants of distribution of waves r 1 © Rustamova M.A., 2019 This work is licensed under a Creative Commons Here, r - radius vector of a particle of a network; p - power of a reaction of the cylinder; Attribution 4.0 International License e1 , e2 - the relative lengthening, corresponding threads; ТЕОРИЯ УПРУГОСТИ 149 Rustamova M.A. Structural Mechanics of Engineering Constructions and Buildings, 2019, 15(2), 149-157 s1 , s2 · Lagranzhevy of co-ordinates of particles of r ¶τ2 = cosγ r ¶i r ¶ (cosγ2 ) o i + threads; σ1 and σ2 the conditional pressure defined ¶s 2 ¶s ¶s r 2 2 2 r as the sum of tension of separate threads of one family (crossing a site of a thread of other family), carried to an initial length of a considered element. Such distribution of weight and efforts is admis- +sinγ2 ¶j ¶s2 ¶ (sinγ2 ) · j . ¶s2 sible at sufficient dense network, ρ1 +ρ2 - weights of elements of the network, having corresponding di- Considering r r r rections on area unit in an initial condition, τ1 , τ2 ¶i = ¶i = 0; ¶j = - sinγ1 k ; individual vectors tangents to threads, n - a normal ¶s1 ¶s2 ¶s1 r r r to a surface of the cylindrical basis. 1. Coordinate system To a basis of a cylindrical rsystem are accepted ¶j sinγ =- 2 k . ¶s2 r From (3) we will get (figure 1):ran individual vector i parallel to a cylin- ¶τ r ¶ (cosγ ) r (sinγ )2 r ¶ (sinγ ) r der axis, j an individual vector of a tangent to crossr section section of the cylinder, k an individual vec- 1 = ¶s1 r 1 i - ¶s1 r 1 k + r r 1 j. ¶s1 r (4) tor perpendicular to the previous ones, x - is the co- ¶τ ¶ (cosγ ) (sinγ )2 ¶ (sinγ ) 2 2 2 2 ordinate in the direction of the axis of the cylinder, y - is the length of an arc of the circumference of the cylinder. Then = ¶s2 ¶s2 i - k + r r r r j. ¶s2 r r r r r r Also considering r = xi + rk we have τ1 =cosγ1 i + sinγ1 j; τ2 =cosγ2 i + sinγ2 j, (2) r r r r where γ1,2 - corners of threads formed with a cylin- ¶r = ¶t ¶x i + r ¶k ¶t ¶t = ¶x ¶t i + rω j ; (5) der axis. r r r r ¶2r ¶t2 or ¶2 x = ¶t2 i + r ¶ω j + rω ¶j ¶t ¶t 2 r 2 r r r ¶ r = ¶ x i + rεj + rω2 k , ¶t2 ¶t2 where ω - angular speed; ε - angular acceleration. Figure 1. Coordinate system Derivatives: 2. The equations of movement of a cylindrical network Having substituted (4) and (5) in (1) we will get r r r ¶ (σ cosγ )i - σ1 (sinγ )2 k + ¶ (σ sinγ ) j + r ¶τ1 = cosγ r ¶i r ¶ (cosγ1 ) · i + ¶s 1 1 r r 1 ¶s 1 1 2 r r 1 1 1 + ¶ · cosγ i - σ2 sinγ k + ¶ · sinγ j = ¶s1 ¶s1 ¶s1 ( ¶s2 2 2 ) ( r 2 ) ( ¶s2 2 2 ) r r 2 r r r r +sinγ ¶j ¶ (sinγ1 ) o j ; (3) = (ρ +ρ ) ¶ x i + (ρ +ρ ) rεj + (ρ +ρ ) rω2k + pn; 1 ¶s1 ¶s1 1 2 ¶t2 1 2 1 2 150 THEORY OF ELASTICITY Рустамова М.А. Строительная механика инженерных конструкций и сооружений. 2019. Т. 15. № 2. С. 149-157 ¶ ¶ ¶2 x where e also γ are values of parameters in an ini- ¶s1 (σ1cosγ1 ) + ¶s2 (σ2cosγ2 ) = (ρ1 +ρ2 ) ; ¶t 2 0 0 tial condition. 1 1 ¶ (σ sinγ ) + ¶ (σ2sinγ2 ) = (ρ1 +ρ2 ) rε; (6) Using (8) of the first equation (6) it is possible to write: ¶s1 ¶s2 ¶σ 1 ¶x ¶ æ 1 ¶x ö ¶2 x · σ1 (sinγ )2 - σ2 (sinγ )2 = (ρ +ρ ) rω2 + p. ¶s 1 + e ¶s + σ ç ÷ = ¶s è 1 + e ¶s ø (ρ1 +ρ2 ) 2 ; ¶t r 1 r 2 1 2 1 ¶σ ¶x 1 ¶e ¶x o ¶2 x Next is the symmetrical arrangement of the right 1+ e ¶s ¶s · σ (1+ e)2 ¶s ¶s + = 1+ e ¶s2 and left fibers. Then the equations (6), considering ¶2 x σ1 = σ2 = σ, γ1 = - γ2 = γ, ω = 0, ε = 0 will become: ; = (ρ1 +ρ2 ) ¶t 2 ¶ ¶2 x σ¢ ¶e ¶x - σ 1 ¶e ¶x + σ ¶ 2 x = 2 (σcosγ) = (ρ1 +ρ2 ) ¶s ; ¶t2 1+ e ¶s ¶s ¶2 x (1+ e)2 ¶s ¶s 1+ e ¶s2 2 σ (sinγ)2 =- p. r (7) = (ρ1 +ρ2 ) ; ¶t2 æ σ¢ σ ö 1 ¶2 x æ ¶x ö2 o ¶2 x Geometrical correlations ç - 1+e (1 ÷ e)2 1 e s2 ç s ÷ + 1 = e s2 ç + ÷ + ¶ è ¶ ø + ¶ è ø Let's define a derivative of a radius-vector r 2 r r = (ρ +ρ ) ¶ x . (11) with respect to s. Having designated r = xi + rk . r r r r 1 2 ¶t 2 ¶r = ¶x i + ¶k r = ¶x i + ¶y j. From (11) we will get the following equation: ¶s ¶s ¶s ¶s ¶s éæ σ¢ σ öæ ¶x ö2 σ ù ¶2 x Where according to (1) and (3) êç êç (1+ e)2 ÷è ø - (1+ e)3 ÷ç ¶s ÷ + ú = 1+ e ú ¶s2 r r (1+ e1 )cosγ1 i + (1+ e1 )sinγ1 j = ¶r ; ëè ø 2 = ( ) ¶ x û (12) ¶s1 r r ¶r ρ1 +ρ2 ¶t2 . (1+ e2 )cosγ2 i + (1+ e2 )sinγ2 j = ; ¶s2 Last equation represents quasilinear equation in partial derivatives. (1+ e)cosγ = ¶x ; (8) æ ö 2 ¶s a = ç σ¢ - σ ÷æ ¶x ö + σ . ç (1+ e)2 è ø (1+ e)3 ÷ç ¶s ÷ 1+ e (1+ e)sinγ = ¶y . ¶s (9) è ø 2 æ¶x ö ¶x As the network does not rotate, then y = const. Here ¶s e = ç ÷ è ø +(1+e0 )sinγ0 -1; ε = ¶s ; ¶ ((1+ e)sinγ) = 0 ¶t ¶s ç ÷ σæ ¶x ö it is set. è ø If we take σ, σ¢ in the following way 2 ¶s ç ÷ or σ = α æ ¶x ö è ø ¶s ç ÷ ; σ¢ = 2αæ ¶x ö , we get the above given è ø (1+e0 )sinγ0 =(1+ e)sinγ, (10) plot. ТЕОРИЯ УПРУГОСТИ 151 Rustamova M.A. Structural Mechanics of Engineering Constructions and Buildings, 2019, 15(2), 149-157 From (8) and (10) 5 2 4 (1+ e)2 cos2 γ = ¶ x ; 3 a(e) 2 ¶s2 0 0 (1+e )2sin2 γ =(1+ e)2sin2 γ; 2 1 ¶ x 2 2 2 0 5 10 15 e + (1+e ) sin γ =(1 + e) . ¶S 2 0 0 From (17) (17) Figure 2. The graph for dependence between quantities ε and a(ε) ¶e ¶x ¶2 x 2(1 + e) = 2 ¶s ¶s ¶s2 or Let's consider another case. Flat nonlinear elastic, in other words, ¶e = . 1 ¶x ¶2 x 2 (18) 2 ¶x æ ¶x ö ¶s (1 + e) ¶s ¶s è ø σ = α1 × ¶s + α2 ç ¶s ÷ , system (8), (9) and (10) can Using (15), (16) and (18) in (14) we will get be reduced to one quasilinear equation of the second order. æ ¶x æ 1 (¶x ) ¶ x 1 ¶ x ö 1 ¶x ¶ x ö 2 2 2 2 1 α ç ç- + ÷+ ÷+ From (6) follows è ¶s è 3 2 2 2 · e ¶s ¶s oe ¶s ø · e ¶s ¶s ø (1 ) 1 1 ; 2 × (σcosγ) = (ρ1 +ρ2 )× 2 (13) 2 ç( ) ç 3 ( ) 2 + 2 2 ÷+ 2 2 ÷ = ¶ ¶2 x +α - æ ¶x 2 æ 2 2 1 ¶x ¶ x 1 ¶ x ö 2 ¶x ¶x ¶ x ö ¶S ¶t è ¶s è (1+ e) ¶s ¶s 1+ e ¶s ø 1+ e ¶s ¶s ¶s ø ¶ æ ¶x 2 æ ¶x ö ö ¶2 x (ρ +ρ ) = 1 2 × ¶2 x ; 2 × çç α1 × + α2 ç ÷ ; ÷÷ cosγ= (ρ1 +ρ2 )× 2 2 2 ¶t ¶S è ¶s è ¶s ø ø ¶t æ 1 3 æ ¶x ö ¶ 2 x 2 ¶x ¶ 2 x ö 2 α1 çç - (1 + e)3 ç ¶s ÷ ¶s 2 + 1 + e ¶s ¶s 2 ÷÷ + ¶ æ ¶x ö ¶ æ æ ¶x ö ö è è ø ø ç 2 × ç α1 × cosγ ÷ + 2 × ç α2 × cosγ ç ÷ ¶S ¶s ¶S ¶s ç ÷÷ = æ 4 2 2 2 ö è ø è è ø ø +α - 1 2 ç æ ¶x ö 3 ç ÷ ¶ x 3 2 æ ¶x ö ç ÷ + ¶ x 2 ÷÷ = ¶2 x è (1 + e) è ¶s ø ¶s 1 + e è ¶s ø ¶s ø = (ρ1 +ρ2 )× ; ¶t 2 = (ρ1 + ρ 2 ) × 2 ¶ 2 x . ¶t 2 ¶x ¶cosγ ¶2 x 2 × α1 ¶S ¶s + 2 × α1cosγ ¶s2 + 2 2 2 æ ¶x ö ¶cosγ ¶x ¶2 x Here, a0 = . ρ1 +ρ2 è ø +2 ×α2 ç ¶s ÷ ¶S ¶2 x · 4 ×α2cosγ ¶s ¶s2 = Last equation can be represented in the above given form: . = (ρ1 +ρ2 ) × 2 (14) 3 2 ¶t é 1 ë - (α + α ¶x )(¶x ) + æ 1 (2α + 3α ¶x )¶x öù ¶ x = ê (1 + e)3 1 2 ¶s ¶s ç è (1 + e) 1 2 ÷ú 2 ¶s ¶s øû ¶s From (8) ( ) 2 ρ +ρ ¶ x = 1 2 × . 2 cosγ = 1 ¶x ; (15) 2 ¶t ¶x ¶ 2 x ¶2 x 1+ e ¶s ¶cosγ 1 ¶e ¶x 1 ¶2 x a æ ö è ø ç ¶s ÷ ¶s2 = a0 × ¶t 2 . (19) =- + ¶S (1+ e)2 ¶s ¶s . (1+ e) ¶s2 (16) The last equation is a quasilinear partial differential equation. 152 THEORY OF ELASTICITY Рустамова М.А. Строительная механика инженерных конструкций и сооружений. 2019. Т. 15. № 2. С. 149-157 The coefficient at ¶x ¶ 2 x ¶s 2 in (19) increases with and t dx s = -adx ; (23) the growth of , therefore speed of waves with ¶s deformation growth increases, conducts to the formation of shock waves [8]. ( é 1 1 2 0 ê 3 ç ¶x )(¶x )3 æ ( ) 1 ¶x 1 2 ¶x öù ÷ú a = a - α +α + 2α + 3α . Continuous waves will occur when unloading a pre-stretched cylinder. Here, too, the method of ë (1+ e) ¶s ¶s è (1+ e) ¶s ¶s øû characteristics is used (figure 3). The front of an unloading wave moves with a speed 0 a(e ). In the field of SОA (figure 2) a rest condition. From a condition on negative characterisxs tic BC follows xt = -òadxs ; differentiating in x 0 s a direction of the positive characteristic we have dx = -adx . t s Comparing with (22) we get xt = const, xs = const . In other words, on positive characteristics xt , xs are constant. From (20) we have, considering a constancy xs on the characteristic x = a(t -t0 ). At x = 0 we choose t0 and define ε. (24) Figure 3. The method of characteristics From a point 0 wave extends with the maximum From (24) 0 t = t - x speed a(e0 ) as waves with smaller deformation exa tend with smaller speed and will not influence a condition at the front. Let the cylinder to locate in the stretched condition e0 . On border the cylinder unloads, in other words, and accordingly J = J0 (t0 ) J = J æ t - x ö. (25) its end moves with a speed of J. 0 ç a ÷ Characteristics of the equation (19) have a form: è ø Let's consider an example: γ = π 0 4 and ds = adt. ds = -adt. (20) (21) π γ0 = , 6 e0 = 0,1; a0 = 5000 м/с. Conditions on characteristics The plot of ( a(xs ) = a(ε)(xs = ε) , f (ε) is dxt s ç s = adx æ ¶x = x , = x ¶x ö t ÷ (22) shown on figure 3 and the plot of manifested on figure 4. p(ε), m(ε) is è ¶s ¶t ø ТЕОРИЯ УПРУГОСТИ 153 Rustamova M.A. Structural Mechanics of Engineering Constructions and Buildings, 2019, 15(2), 149-157 f(e) a( k) 8.8´104 8.625´104 8.45´104 8.275´104 8.1´104 7.925´104 7.75´104 7.575´104 7.4´104 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 e , k Figure. 4 The graph for dependence between quantities ε, k and f (ε), a(k ) : æ γ = π ; e = 0, 1 ; a = 5000 м/с ö ç ÷ 4 è 0 0 0 ø 0.11 0.096 0.083 m(e) p(k) 0.069 0.055 0.041 0.028 0.014 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 e , k Figure. 5. The graph for dependence between quantities ε, k and f (ε), a(k ) : æ γ = π ; e = 0, 1 ; a = 5000 м/с ö ç ÷ 6 è 0 0 0 ø Let the cylinder on border s = 0 unload with The equation (26) is the equation for defining an a speed υ(t). From (25) axial deformation of a network formation of fibers e). xs = ε (unlike de- ε J(t) = -ò a(xs )dxs . (26) Approximately having presented integral (19) in the form of the sum, we have: ε ε0 Where J is a function of the top limit of an integral. J = -ò a(η)dη; ε0 (27) 154 THEORY OF ELASTICITY Рустамова М.А. Строительная механика инженерных конструкций и сооружений. 2019. Т. 15. № 2. С. 149-157 ε0 J = òa(η)dη ; or J = f (ε) ε inverse relationship ε ®J on border. As positive J0 = a(ε0 )Dε; J1 = (a(ε0 ) + a(ε1 )) × Dε; J2 = (a(ε0 ) + a(ε1 ) + a(ε2 )) × Dε; ........................................... characteristics are rectilinear, it is possible to define ε in all area of movement. Functional dependence of speed of movement - speed of a wave and deformation for the given example is presented in the tables 1 and 2. Jn = (a(ε0 ) + a(ε1 ) + ××× + a(εn ))× Dε Jn = f (ε0 -ε). (28) π 0 Calculated values of the utilized parameters (for γ = ) 4 Table 1 ε0 ε1 ε2 ε3 ε4 ε5 ε6 ε7 ε8 ε9 ε10 ε11 0.778 0.770 0.762 0.754 0.746 0.738 0.730 0.722 0.714 0.706 0.698 0.690 e(ε0) e(ε1) e(ε2) e(ε3) e(ε4) e(ε5) e(ε6) e(ε7) e(ε8) e(ε9) e(ε10) e(ε11) 0.1 0.094 0.089 0.083 0.078 0.072 0.067 0.061 0.056 0.050 0.045 0.040 ϑ0 ϑ1 ϑ2 ϑ3 ϑ4 ϑ5 ϑ6 ϑ7 ϑ8 ϑ9 ϑ10 ϑ11 747·103 1.493·103 2.237·103 2.980·103 3.721·103 4.461·103 5.199·103 5.936·103 6.672·103 7.406·103 8.139·103 8.87·103 a(ε0) a(ε1) a(ε2) a(ε3) a(ε4) a(ε5) a(ε6) a(ε7) a(ε8) a(ε9) a(ε10) a(ε11) 7.753×104 7.732×104 7.711×104 7.689×104 7.667 ×104 7.644×104 7.621×104 7.598×104 7.575×104 7.551×104 7.526×104 7.502×104 π Calculated values of the utilized parameters (for γ0 = ) 6 Table 2 ε0 ε1 ε2 ε3 ε4 ε5 ε6 ε7 ε8 ε9 ε10 ε11 0.953 0.950 0.947 0.944 0.941 0.938 0.935 0.932 0.929 0.926 0.923 0.920 e(ε0) e(ε1) e(ε2) e(ε3) e(ε4) e(ε5) e(ε6) e(ε7) e(ε8) e(ε9) e(ε10) e(ε11) 0.230 0.228 0.225 0.223 0.221 0.219 0.216 0.214 0.212 0.209 0.207 0.205 ϑ0 ϑ1 ϑ2 ϑ3 ϑ4 ϑ5 ϑ6 ϑ7 ϑ8 ϑ9 ϑ10 ϑ11 866.7 1.732·103 2.597·103 3.461·103 4.324·103 5.186·103 6.048·103 6.908·103 7.767·103 8.626·103 9.483·103 1.034·103 a(ε0) a(ε1) a(ε2) a(ε3) a(ε4) a(ε5) a(ε6) a(ε7) a(ε8) a(ε9) a(ε10) a(ε11) 8.667 ×104 8.658 ×104 8.648 ×104 8.639×104 8.630×104 8.621×104 8.612×104 8.603×104 8.593×104 8.584×104 8.575 ×104 8.566×104 Сonclusions Setting on border speed of movement of the end of a network as a time function it is possible to define deformation as time function on the end of a network and to the above-stated form everywhere in area SOt. Depending on distribution of speed on the border, deformation of a constant on characteristics is defined (figures 4 and 5).

About the authors

Mexseti A Rustamova

Institute of Mathematics and Mechanics of National Academy of Sciences of Azerbaijan

Author for correspondence.
Email: mehsetir@gmail.com

PhD in Physical and Mathematical Sciences, Leading Researcher, Associate Professor, Department of Wave Dynamics, Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan

9 B. Vahabzadeh St., Baku, AZ 1143, Republic of Azerbaijan

References