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Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin's model
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ARTICLE IN PRESS Journal of Applied Mathematics and Mechanics xxx (2016) xxx–xxx
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Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin’s model夽 Yu.A. Chirkunov Novosibirsk State Technical University, Novosibirsk, Russia
a r t i c l e
i n f o
Article history: Received 10 November 2014 Available online xxx
a b s t r a c t Non-linear longitudinal oscillations of a viscoelastic rod are investigated in Kelvin’s model. All the secondorder conservation laws for the differential equation describing these oscillations see obtained, which, using non-local variables, generate two non-linear systems of differential equations, equivalent to this equation. A group analysis of these systems is carried out. All their essentially different invariant solutions (unconnected with point transformations) are obtained, which are either found in explicit form or the search for them is reduced to solving non-linear integro-differential equations, which opens up new possibilities for analytical and numerical investigations. The presence of arbitrary constants in these equations enables them to be used to investigate various boundary value problems. With additional conditions, the existence and uniqueness of the solutions of certain boundary value problems, describing non-linear longitudinal oscillations of a visco-elastic rod in Kelvin’s model, are established. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Non-linear longitudinal oscillations of a visco-elastic rod in Kelvin’s model are described by the equation (Ref. 1, p. 707, Ref 2, p. 575) (1.1) where x is a coordinate characterizing the position of the transverse section of the rod, u = u(t, x) is the longitudinal displacement of the section of the rod in a time t, = / 0 is the viscosity coefficient, = (ux ) is the stress and ’(ux ) > 0, where the prime denotes differentiation with respect to ux . It is assumed that the condition ”(ux ) = / 0, which indicates the non-linearity of the rod oscillations, is satisfied. Without loss of generality we can assume that = 1; this can, in fact, be obtained using an extension transformation of the variables t, x and u. Henceforth we will consider Eq. (1.1) with = 1. The linearized version of Eq. (1.1) is employed in creep theory3 when investigating the linear deformation of a simple visco-elastic body. Non-linear deformation has already been described by Eq. (1.1).4 Existence and uniqueness solutions of Cauchy’s problem have been obtained for Eq. (1.1), and the problem of their stability to initial perturbations has been investigated.1,2 In this paper, using the second-order conservation laws obtained for Eq. (1.1), we obtain two non-linear systems, equivalent to this equation. The group analysis of these systems opens up new possibilities for the analytical and numerical investigation of Eq. (1.1). Existence and uniqueness solutions of some boundary-value problems are established. 2. Systems of equations, equivalent to Eq. (1.1), generated by the conservation laws The second-order conservation laws for Eq. (1.1) are defined by the vector A = (A0 , A1 ), the components of which are functions of the variables t, x, u, ut , ux , utt , utx , uxx , and in view of Eq. (1.1), satisfy the relation5–7 (2.1)
夽 Prikl. Mat. Mekh., Vol. 79, No. 5, pp. 717–727, 2015. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jappmathmech.2016.03.012 0021-8928/© 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Chirkunov YuA. Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin’s model. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.03.012
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where Dt and Dx are operators of total differentiation with respect to t and x respectively. As a result of splitting Eq. (2.1) into parametric derivative functions of u, we obtain the following system of governing equations:
(2.2) After the third extension of system (2.2), the following equations are added to it:
After the fourth extension of system (2.2) the equation A0uxx uxx = 0 is added to it, by virtue of which the system is in involution and is integrated. The general solution of this system has the form
where c1 and c2 are arbitrary constants, and = (t, x, u, ut , ux ) is an arbitrary function, which gives trivial conservation laws.5 Second-order non-trivial conservation laws5–7 for Eq. (1.1) are generated by the following two conservation laws: (2.3) (2.4) Each of these defines a non-local variable, by means of which Eq. (1.1) can be written in the form of the equivalent system of equations (A) (B) Here w = w(t, x) and v = v(t, x) are non-local variables. For any solution (u, w) of system (A) and for any solution (u, v) of system (B) the function u = u(t, x) is a solution of Eq. (1.1). The inverse assertion also holds: for any solution u = u(t, x) of Eq. (1.1), functions w = w(t, x) and v = v(t, x) are obtained, such that (u, w) and (u, v) are solutions of system (A) and system (B) respectively. A new method for the group classification of systems of differential equations was proposed in Ref. 8, based on the use of generalized equivalence transformations, and a group classification was carried out with respect to the stress (ux ) of systems of equations (A) and (B). These results are presented below. 3. The results of group classification of system (A) and system (B) The four-parameter (five-parameter) Lie group of transformations, generated by the operators
is a series of fundamental groups of system (A) (system (B)), i.e., this system is allowed for an arbitrary element (ux ). Expansion of the series is only possible with respect to one operator. Namely, system (A) (system (B)) allows of the additional operator
if (3.1) the additional operator
Please cite this article in press as: Chirkunov YuA. Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin’s model. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.03.012
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if (3.2) and the additional operator
if (3.3) where ␥ is an arbitrary constant. 4. Formulae for producing the solutions Integration of the Lie equations9,10 for the operators Y1 , Y2 , Y3 and Y4 (the operators Z1 , Z2 , Z3 , Z4 and Z5 ) gives the following formulae for producing solutions of non-linear system (A) (system (B)): if u = u(t, x) and w = w(t, x) are a solution of system (A) (u = u(t, x), v = v(t, x) is a solution of system (B)), the functions
(4.1)
(4.1)’ t , x , u , w (t , x , u , )
is a solution of this system, where a1 , . . ., a4 (b1 , . . ., b5 ) are arbitrary constants. written for the variables In cases (3.1)–(3.3), the operators Y5 , Y6 and Y7 (the operators Z6 , Z7 and Z8 ) generate additional formulae for producing solutions. Thus, for any solution u = u(t, x), w = w(t, x) of system (A) (solutions u = u(t, x), v = v(t, x) of system (B)) the following functions are a solution of this system, written for the variables t , x , u , w (t , x , u , ): for condition (3.1)
(4.2)
(4.2)’ for condition (3.2)
(4.3)
(4.3)’ for condition (3.3)
(4.4)
(4.4)’ where a5 , a6 , a7 , (b6 , b7 , b8 ) are arbitrary positive constants. 5. Invariant and partially invariant solutions of systems (A) and (B) In the solutions of systems (A) and (B) only the component u = u(t, x) – the longitudinal displacement of the section of the rod, has a physical meaning. The non-local variables w and v play an auxiliary role: they are used to find the displacement u. Hence, we will henceforth only present this part of the solution of systems (A) and (B). It can be verified directly that both systems for any stress (ux ) have a solution in which the displacement has the form
Please cite this article in press as: Chirkunov YuA. Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin’s model. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.03.012
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where c1 ,. . ., c5 are arbitrary constants. We will regard this displacement as trivial, while the solution with the trivial displacement will be regarded as a trivial solution of the system. The following assertions can be established directly.
Theorem 1. Suppose H is an arbitrary subgroup of the Lie group of transformations Y1 , Y2 , Y3 , Y4 . Then, for any element / 0) all the invariant and partially invariant H-solutions of system (A) are its trivial solutions. (ux )( ’ ’ (ux ) =
Theorem 2. Suppose H is an arbitrary subgroup of the Lie group of transformations Z1 , Z2 , Z3 , Z4 , Z5 . Then, apart from transformations / 0) only three essentially different (unconnected by point transformations) non-trivial invariant (4.1)’ for any element (ux ) ( ’ ’ (ux ) = H-solutions of system (B) exist. These are solutions invariant with respect to the subgroups (5.1) where ␣ and  are arbitrary constants. All the partially invariant H-solutions of this system are either reduced to its invariant solutions or are trivial solutions. For the solutions of system (B), invariant with respect to subgroup (5.1), indicated in Theorem 2: a, b and c are non-trivial longitudinal displacements of the rod section respectively, such that
(5.2) where P() is the solution of the integral equation
(5.3)
(5.4)
(5.5) where P() is the solution of the integral equation
(5.6) Here ␣, , c1 , c2 and 0 are arbitrary constants. Formulae (5.2), (5.3) and formulae (5.4), (5.6) with  = 0 define a travelling wave, and when  = / 0 it defines extensions of a travelling / 0 it defines its dynamic extension. The presence in wave. Solution (5.4) with  = 0 describes static deformation of the rod, and when  = these formulae of arbitrary constants, and also formulae of the production of solutions (4.1)’ enables us to obtain solutions of Eq. (1.1), which satisfy different boundary conditions. By virtue of Theorems 1 and 2, the nontrivial invariant and partially invariant H-solutions of system (A), and also the H-solutions of may exist only if the stress (ux ) is expressed in terms system (B), which differ from solutions (5.2) – (5.6) of the deformation as given by formulae (3.1) – (3.3), while the subgroup H ∈ / Y1 , Y2 , Y3 , Y4 for system (A) and H ∈ / Z1 , Z2 , Z3 , Z4 , Z5 for system (B). The result of an analysis of the invariant H-solutions of system (A) with condition (3.1) is presented in the following theorem. (A), unique apart from transformations (4.1) and (4.2) with condition (3.1) is a Theorem 3. The nontrivial invariant H-solutionof system solution, invariant with respect to the subgroup Y5 . It has the form (5.7) Hence, and everywhere henceforth, we use the notation
Substitution into system (A) gives the factor system
Elimination of W from this factor system leads to the equation (5.8) Please cite this article in press as: Chirkunov YuA. Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin’s model. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.03.012
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where c1 is an arbitrary constant. The introduction of the new unknown function
enables us to write Eq. (6.8) in the form of the following system, equivalent to it,
(5.9) from which the function U() is defined by the quadrature
(5.10) where 0 > 0 and c2 are arbitrary constants, while the function P() is the solution of the integro-differential equation
(5.11) The solution, defined by the first formula of (5.7) with the function U(), given by formulae (5.10) and (5.11), describes longitudinal oscillations of a viscoelastic rod, the elastic properties of which are defined by the relation = exp(ux ), while on its end x = x0 > 0 at the initial instant of time t = t0 > 0, the displacement, velocity and acceleration are specified: (5.12) where u0 , u0 and u0 are arbitrary constants. For this solution the quantity (5.13) maintains a constant value along the trajectories t = kx2 (k = cost > 0). The constants c1 and c2 then take the form
(5.14)
(5.15) The following initial conditions are added to system (5.9):
(5.16) while the last initial condition (5.16) is added to integro-differential equation (5.11). In view of the smoothness of the right-hand sides of system (5.9), the solution of Cauchy’s problem (5.9) and (5.16) exists is unique, and, consequently, a unique solution of integro-differential equation (5.11) exists with the last initial condition of (5.16). Hence, moreover, a unique solution of Eq. (1.1) exists for = exp(ux ), which satisfies conditions (5.12), for which the quantity (5.3) maintains a constant value along the trajectories t = kx2 (k = const > 0). This solution is determined by formulae (5.7), (5.10), (5.11), (5.13) and (5.14) – (5.16). We can take other conditions instead of conditions (5.12). For example, at the end of the rod x = x0 > 0, at the initial instant of time t = t0 > 0, the following quantities may be specified, in addition to the displacement (the first equation of (5.12): 1) 2) 3) 4) 5)
the velocity and rate of deformation: ut (t0 , x0 ) = u0 , utx (t0 , x0 ) = u0 the rate and gradient of the deformation: ut (t0 , x0 ) = u0 , uxx (t0 , x0 ) = u0 the deformation and acceleration: ux (t0 , x0 ) = u0 , utt (t0 , x0 ) = u0 the deformation and rate of deformation: ux (t0 , x0 ) = u0 , utx (t0 , x0 ) = u0 the deformation and deformation gradient: ux (t0 , x0 ) = u0 , uxx (t0 , x0 ) = u0
In all cases moreover a unique solution of Eq. (1.1) exists for = exp(ux ), which satisfies the first condition of (5.12) and conditions from the sets 1 – 5, for which the quantity (5.13) retains a constant value along the trajectory t = kx2 (k = const > 0). This solution is determined from formulae (5.7), (5.10), (5.11) and (5.15). The last initial condition (5.16) for cases 1 and 2 remains, while for cases 3, 4 and 5 it is replaced by the condition
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Equality (5.14) in cases 1 – 5 is replaced by the following equalities respectively:
Here
The result of the analysis of the invariant H-solutions of system (A) with condition (3.1) is presented in the following theorem. Theorem 4. The set of non-trivial invariant longitudinal displacements of the section of the rod, defined by system (B) with condition (3.1), apart from transformations (4.1)’ and (4.2)’ is exhausted by the displacements specified by formulae (5.2)–(5.7), (5.10) and (5.11). The results of an analysis of the invariant H-solutions of system (A) with condition (3.2) is presented in the following theorem. (A) with condition (3.2), unique apart from transformations (4.1) and (4.3), is Theorem 5. The non-trivial invariant H-solutionof system a solution invariant with respect to the subgroup Y6 . It has the form
(5.17) Like solution (6.7) it is established that
(5.18) while the function P() is the solution of the integro-differential equation
(5.19) where 0 > 0 and c1 and c2 are arbitrary constants. The solution, defined by the first formula of (5.17), with the function U(), specified by formulae (5.18) and (5.19), describes longitudinal oscillations of a viscoelastic rod, the elastic properties of which are defined by the relation = ln ux , while at its end x = x0 > 0 at the initial instant of time t = t0 > 0 one of the six sets of conditions: (5.12) and 1 – 5 is specified. For this solution the quantity u(t, x)/x3 keeps a constant −3/2 value along the trajectory t = kx2 (k = const > 0). For all six cases c1 = t0 u0 , but the expressions for the constant c2 are lengthy and are not given here. We add the following initial condition to integro-differential equation (5.19):
and
The result of an analysis of the invariant H-solutions of system (B) with condition (3.2) is presented in the following theorem. Theorem 6. The set of non-trivial invariant longitudinal displacements of the section of the rod, defined by system (B) with condition (3.2), apart from transformations (4.1)’ and (4.3)’ is exhausted by the displacements specified by formulae (5.2) – (5.6) and (5.17) and (5.19). / 1/2) is presented in the following theorem. The result of an analysis of the invariant H-solutions of system (A) with condition (3.3) (␥ = (A),apart from transformations (4.1) and (4.4), with condition (3.3) Theorem 7. The unique non-trivial invariant H-solution of system (␥ = / 1/2), is a solution invariant with respect to the subgroup Y7 + ␣Y4 , where ␣ is an arbitrary constant. It has the form
Please cite this article in press as: Chirkunov YuA. Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin’s model. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.03.012
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(5.20) Like solution (5.7) it is established that
(5.21) while the function P() is the solution of the integro-differential equation
(5.22) where 0 > 0 and c1 and c2 are arbitrary constants. The solution u = x2␥−1 U() with the function U(), given by formulae (5.17) and (5.18), describes longitudinal oscillations of a viscoelastic rod, the elastic properties of which are given by the relation
while on its end x = x0 > 0 at the initial instant of time t = t0 > 0 one of the six sets of conditions is specified: (5.12) and 1 – 5. For this solution 1/2−␥ the quantity x1−2␥ u(t, x) retains a constant value along the trajectory t = kx2 (k = const > 0). For all six cases c1 = t0 u0 . The initial condition for integro-differential equation (5.22), has the form
and
The expressions for the quantity c2 in these cases are not given here in view of their length. The result of an analysis of the invariant H-solutions of system (B) with condition (3.3) (␥ = / 1/2) is presented in the following theorem. Theorem 8. The set of non-trivial invariant longitudinal displacements of the section of the rod, defined by system (B) with condition (3.3) (␥ = / 1/2), apart from transformations (4.1)’ and (4.4)’ is exhausted by the displacements specified by formulae (5.2) – (5.6) and (5.20) – (5.22). With condition (3.3) (␥ = 1/2) the stress depends on the deformation as given by the relation = u3x . An additional operator for system (A) (system (B)) is the operator Y7 (operator Z8 ) with ␥ = 1/2, while the additional formula for producing the solutions is formula (4.4) (formula (4.4)’). However, by comparison with case (3.3) (␥ = / 1/2) the optimum system of subgroups of the main group of both system (A) and system (B) is changed. An analysis of the invariant H-solutions of system (A) with = u3x is presented in the following theorem. Theorem 9. The unique non-trivial invariant H-solution of system (A) with = u3x , apart from transformations (4.1) and (4.4), is the solution invariant with respect to the subgroup Y7 + ␣Y3 (␥ = 1/2), where ␣ is an arbitrary constant. It has the form (5.23) Like solution (5.7) we have
(5.24) while the function P() is the solution of integro-differential equation
(5.25) where 0 > 0, ␣, c1 and c2 are arbitrary constants. The solution defined by the first formula of (5.23), with function U(), specified by formulae (5.24) and (5.25), describes longitudinal oscillations of a viscoelastic rod, whose elastic properties are defined by the relation = u3x , while on its ends x = x0 > 0 at the initial instant of time t = t0 > 0 one of the six sets of conditions (5.12), 1 – 5 are given. For this solution the quantity u(t, x) – ␣ln x retains a constant value Please cite this article in press as: Chirkunov YuA. Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin’s model. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.03.012
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along the trajectories t = kx2 (k = const > 0). In all six cases c1 = u0 − condition
␣ 2
ln t0 . We must add to integro-differential equation (5.25) the initial
and
Expressions for the quantity c2 in these cases are not given here in view of their length. An analysis of the invariant H-solutions of system (B) for = u3x is presented in the following theorem. Theorem 10. The set of non-trivial invariant longitudinal displacements of the section of the rod, defined by system (B) with = u3x , apart from transformations (4.1)’ and (4.4)’ is covered by the variables specified by formulae (5.2) – (5.6), (5.23) – (5.25). Acknowledgements This research was supported financially by the Ministry of Education and Sconce of the Russian Federation (2014/138, No. 435), and the Program for the Support of the Leading Scientific Schools (NSh-2133.2014.1) and the Russian Foundation for Basic Research (12-01-00648). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Greenberg JM, MacCamy RC, Mizel VJ. On the existence, uniqueness, and stability of solutions of the equation ␦’ (ux )uxx + uxtx = 0 utt . J. Math. Mech 1968;17(7):707–28. Greenberg JM. On the existence, uniquencess, and stablity of solutions of the equation 0 ¯ x ) ¯ xx + ¯ xxt . J. Math. Analys. Appl 1969;(25):575–91. ¯ tt = E( Rzhanitsyn AR. Creep Theory. Moscow: Izd Lit Stroitel; 1968. Germain P. Cours de mecanique des milieu continus. T. 1. Toeorie generale. Paris: Masson; 1973. p. 350. Olver PJ. Applications of Lie Groups to Differential Equations. N.Y.: Springer; 1986. p. 570. Chirkunov YuA. On group properties and conservation laws for second-order quasi-linear differential equations. J Appl Mech Tech Phys 2009;50(3):408–13. Chirkunov YuA. Method of A-operators and conservation laws for the equations of gas dynamics. J Appl Mech Tech Phys 2009;50(2):213–9. Chirkunov YuA. Generalized equivalence transformations and group classification of systems of differential equations. J Appl Mech Tech Phys 2012;53(2):147–55. Chirkunov YuA, Khabirov SV. Elements of Symmetrical Analysis of the Differential Equations of Continuum Mechanics. Novosibirsk: Izd NGTU; 2012. Ovsyannikov LV. Group Analysis of Differential Equations. Moscow: Nauka; 1978.
Translated by R.C.G.
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ARTICLE IN PRESS Journal of Applied Mathematics and Mechanics xxx (2016) xxx–xxx
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin’s model夽 Yu.A. Chirkunov Novosibirsk State Technical University, Novosibirsk, Russia
a r t i c l e
i n f o
Article history: Received 10 November 2014 Available online xxx
a b s t r a c t Non-linear longitudinal oscillations of a viscoelastic rod are investigated in Kelvin’s model. All the secondorder conservation laws for the differential equation describing these oscillations see obtained, which, using non-local variables, generate two non-linear systems of differential equations, equivalent to this equation. A group analysis of these systems is carried out. All their essentially different invariant solutions (unconnected with point transformations) are obtained, which are either found in explicit form or the search for them is reduced to solving non-linear integro-differential equations, which opens up new possibilities for analytical and numerical investigations. The presence of arbitrary constants in these equations enables them to be used to investigate various boundary value problems. With additional conditions, the existence and uniqueness of the solutions of certain boundary value problems, describing non-linear longitudinal oscillations of a visco-elastic rod in Kelvin’s model, are established. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Non-linear longitudinal oscillations of a visco-elastic rod in Kelvin’s model are described by the equation (Ref. 1, p. 707, Ref 2, p. 575) (1.1) where x is a coordinate characterizing the position of the transverse section of the rod, u = u(t, x) is the longitudinal displacement of the section of the rod in a time t, = / 0 is the viscosity coefficient, = (ux ) is the stress and ’(ux ) > 0, where the prime denotes differentiation with respect to ux . It is assumed that the condition ”(ux ) = / 0, which indicates the non-linearity of the rod oscillations, is satisfied. Without loss of generality we can assume that = 1; this can, in fact, be obtained using an extension transformation of the variables t, x and u. Henceforth we will consider Eq. (1.1) with = 1. The linearized version of Eq. (1.1) is employed in creep theory3 when investigating the linear deformation of a simple visco-elastic body. Non-linear deformation has already been described by Eq. (1.1).4 Existence and uniqueness solutions of Cauchy’s problem have been obtained for Eq. (1.1), and the problem of their stability to initial perturbations has been investigated.1,2 In this paper, using the second-order conservation laws obtained for Eq. (1.1), we obtain two non-linear systems, equivalent to this equation. The group analysis of these systems opens up new possibilities for the analytical and numerical investigation of Eq. (1.1). Existence and uniqueness solutions of some boundary-value problems are established. 2. Systems of equations, equivalent to Eq. (1.1), generated by the conservation laws The second-order conservation laws for Eq. (1.1) are defined by the vector A = (A0 , A1 ), the components of which are functions of the variables t, x, u, ut , ux , utt , utx , uxx , and in view of Eq. (1.1), satisfy the relation5–7 (2.1)
夽 Prikl. Mat. Mekh., Vol. 79, No. 5, pp. 717–727, 2015. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jappmathmech.2016.03.012 0021-8928/© 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Chirkunov YuA. Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin’s model. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.03.012
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where Dt and Dx are operators of total differentiation with respect to t and x respectively. As a result of splitting Eq. (2.1) into parametric derivative functions of u, we obtain the following system of governing equations:
(2.2) After the third extension of system (2.2), the following equations are added to it:
After the fourth extension of system (2.2) the equation A0uxx uxx = 0 is added to it, by virtue of which the system is in involution and is integrated. The general solution of this system has the form
where c1 and c2 are arbitrary constants, and = (t, x, u, ut , ux ) is an arbitrary function, which gives trivial conservation laws.5 Second-order non-trivial conservation laws5–7 for Eq. (1.1) are generated by the following two conservation laws: (2.3) (2.4) Each of these defines a non-local variable, by means of which Eq. (1.1) can be written in the form of the equivalent system of equations (A) (B) Here w = w(t, x) and v = v(t, x) are non-local variables. For any solution (u, w) of system (A) and for any solution (u, v) of system (B) the function u = u(t, x) is a solution of Eq. (1.1). The inverse assertion also holds: for any solution u = u(t, x) of Eq. (1.1), functions w = w(t, x) and v = v(t, x) are obtained, such that (u, w) and (u, v) are solutions of system (A) and system (B) respectively. A new method for the group classification of systems of differential equations was proposed in Ref. 8, based on the use of generalized equivalence transformations, and a group classification was carried out with respect to the stress (ux ) of systems of equations (A) and (B). These results are presented below. 3. The results of group classification of system (A) and system (B) The four-parameter (five-parameter) Lie group of transformations, generated by the operators
is a series of fundamental groups of system (A) (system (B)), i.e., this system is allowed for an arbitrary element (ux ). Expansion of the series is only possible with respect to one operator. Namely, system (A) (system (B)) allows of the additional operator
if (3.1) the additional operator
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if (3.2) and the additional operator
if (3.3) where ␥ is an arbitrary constant. 4. Formulae for producing the solutions Integration of the Lie equations9,10 for the operators Y1 , Y2 , Y3 and Y4 (the operators Z1 , Z2 , Z3 , Z4 and Z5 ) gives the following formulae for producing solutions of non-linear system (A) (system (B)): if u = u(t, x) and w = w(t, x) are a solution of system (A) (u = u(t, x), v = v(t, x) is a solution of system (B)), the functions
(4.1)
(4.1)’ t , x , u , w (t , x , u , )
is a solution of this system, where a1 , . . ., a4 (b1 , . . ., b5 ) are arbitrary constants. written for the variables In cases (3.1)–(3.3), the operators Y5 , Y6 and Y7 (the operators Z6 , Z7 and Z8 ) generate additional formulae for producing solutions. Thus, for any solution u = u(t, x), w = w(t, x) of system (A) (solutions u = u(t, x), v = v(t, x) of system (B)) the following functions are a solution of this system, written for the variables t , x , u , w (t , x , u , ): for condition (3.1)
(4.2)
(4.2)’ for condition (3.2)
(4.3)
(4.3)’ for condition (3.3)
(4.4)
(4.4)’ where a5 , a6 , a7 , (b6 , b7 , b8 ) are arbitrary positive constants. 5. Invariant and partially invariant solutions of systems (A) and (B) In the solutions of systems (A) and (B) only the component u = u(t, x) – the longitudinal displacement of the section of the rod, has a physical meaning. The non-local variables w and v play an auxiliary role: they are used to find the displacement u. Hence, we will henceforth only present this part of the solution of systems (A) and (B). It can be verified directly that both systems for any stress (ux ) have a solution in which the displacement has the form
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where c1 ,. . ., c5 are arbitrary constants. We will regard this displacement as trivial, while the solution with the trivial displacement will be regarded as a trivial solution of the system. The following assertions can be established directly.
Theorem 1. Suppose H is an arbitrary subgroup of the Lie group of transformations Y1 , Y2 , Y3 , Y4 . Then, for any element / 0) all the invariant and partially invariant H-solutions of system (A) are its trivial solutions. (ux )( ’ ’ (ux ) =
Theorem 2. Suppose H is an arbitrary subgroup of the Lie group of transformations Z1 , Z2 , Z3 , Z4 , Z5 . Then, apart from transformations / 0) only three essentially different (unconnected by point transformations) non-trivial invariant (4.1)’ for any element (ux ) ( ’ ’ (ux ) = H-solutions of system (B) exist. These are solutions invariant with respect to the subgroups (5.1) where ␣ and  are arbitrary constants. All the partially invariant H-solutions of this system are either reduced to its invariant solutions or are trivial solutions. For the solutions of system (B), invariant with respect to subgroup (5.1), indicated in Theorem 2: a, b and c are non-trivial longitudinal displacements of the rod section respectively, such that
(5.2) where P() is the solution of the integral equation
(5.3)
(5.4)
(5.5) where P() is the solution of the integral equation
(5.6) Here ␣, , c1 , c2 and 0 are arbitrary constants. Formulae (5.2), (5.3) and formulae (5.4), (5.6) with  = 0 define a travelling wave, and when  = / 0 it defines extensions of a travelling / 0 it defines its dynamic extension. The presence in wave. Solution (5.4) with  = 0 describes static deformation of the rod, and when  = these formulae of arbitrary constants, and also formulae of the production of solutions (4.1)’ enables us to obtain solutions of Eq. (1.1), which satisfy different boundary conditions. By virtue of Theorems 1 and 2, the nontrivial invariant and partially invariant H-solutions of system (A), and also the H-solutions of may exist only if the stress (ux ) is expressed in terms system (B), which differ from solutions (5.2) – (5.6) of the deformation as given by formulae (3.1) – (3.3), while the subgroup H ∈ / Y1 , Y2 , Y3 , Y4 for system (A) and H ∈ / Z1 , Z2 , Z3 , Z4 , Z5 for system (B). The result of an analysis of the invariant H-solutions of system (A) with condition (3.1) is presented in the following theorem. (A), unique apart from transformations (4.1) and (4.2) with condition (3.1) is a Theorem 3. The nontrivial invariant H-solutionof system solution, invariant with respect to the subgroup Y5 . It has the form (5.7) Hence, and everywhere henceforth, we use the notation
Substitution into system (A) gives the factor system
Elimination of W from this factor system leads to the equation (5.8) Please cite this article in press as: Chirkunov YuA. Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin’s model. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.03.012
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where c1 is an arbitrary constant. The introduction of the new unknown function
enables us to write Eq. (6.8) in the form of the following system, equivalent to it,
(5.9) from which the function U() is defined by the quadrature
(5.10) where 0 > 0 and c2 are arbitrary constants, while the function P() is the solution of the integro-differential equation
(5.11) The solution, defined by the first formula of (5.7) with the function U(), given by formulae (5.10) and (5.11), describes longitudinal oscillations of a viscoelastic rod, the elastic properties of which are defined by the relation = exp(ux ), while on its end x = x0 > 0 at the initial instant of time t = t0 > 0, the displacement, velocity and acceleration are specified: (5.12) where u0 , u0 and u0 are arbitrary constants. For this solution the quantity (5.13) maintains a constant value along the trajectories t = kx2 (k = cost > 0). The constants c1 and c2 then take the form
(5.14)
(5.15) The following initial conditions are added to system (5.9):
(5.16) while the last initial condition (5.16) is added to integro-differential equation (5.11). In view of the smoothness of the right-hand sides of system (5.9), the solution of Cauchy’s problem (5.9) and (5.16) exists is unique, and, consequently, a unique solution of integro-differential equation (5.11) exists with the last initial condition of (5.16). Hence, moreover, a unique solution of Eq. (1.1) exists for = exp(ux ), which satisfies conditions (5.12), for which the quantity (5.3) maintains a constant value along the trajectories t = kx2 (k = const > 0). This solution is determined by formulae (5.7), (5.10), (5.11), (5.13) and (5.14) – (5.16). We can take other conditions instead of conditions (5.12). For example, at the end of the rod x = x0 > 0, at the initial instant of time t = t0 > 0, the following quantities may be specified, in addition to the displacement (the first equation of (5.12): 1) 2) 3) 4) 5)
the velocity and rate of deformation: ut (t0 , x0 ) = u0 , utx (t0 , x0 ) = u0 the rate and gradient of the deformation: ut (t0 , x0 ) = u0 , uxx (t0 , x0 ) = u0 the deformation and acceleration: ux (t0 , x0 ) = u0 , utt (t0 , x0 ) = u0 the deformation and rate of deformation: ux (t0 , x0 ) = u0 , utx (t0 , x0 ) = u0 the deformation and deformation gradient: ux (t0 , x0 ) = u0 , uxx (t0 , x0 ) = u0
In all cases moreover a unique solution of Eq. (1.1) exists for = exp(ux ), which satisfies the first condition of (5.12) and conditions from the sets 1 – 5, for which the quantity (5.13) retains a constant value along the trajectory t = kx2 (k = const > 0). This solution is determined from formulae (5.7), (5.10), (5.11) and (5.15). The last initial condition (5.16) for cases 1 and 2 remains, while for cases 3, 4 and 5 it is replaced by the condition
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Equality (5.14) in cases 1 – 5 is replaced by the following equalities respectively:
Here
The result of the analysis of the invariant H-solutions of system (A) with condition (3.1) is presented in the following theorem. Theorem 4. The set of non-trivial invariant longitudinal displacements of the section of the rod, defined by system (B) with condition (3.1), apart from transformations (4.1)’ and (4.2)’ is exhausted by the displacements specified by formulae (5.2)–(5.7), (5.10) and (5.11). The results of an analysis of the invariant H-solutions of system (A) with condition (3.2) is presented in the following theorem. (A) with condition (3.2), unique apart from transformations (4.1) and (4.3), is Theorem 5. The non-trivial invariant H-solutionof system a solution invariant with respect to the subgroup Y6 . It has the form
(5.17) Like solution (6.7) it is established that
(5.18) while the function P() is the solution of the integro-differential equation
(5.19) where 0 > 0 and c1 and c2 are arbitrary constants. The solution, defined by the first formula of (5.17), with the function U(), specified by formulae (5.18) and (5.19), describes longitudinal oscillations of a viscoelastic rod, the elastic properties of which are defined by the relation = ln ux , while at its end x = x0 > 0 at the initial instant of time t = t0 > 0 one of the six sets of conditions: (5.12) and 1 – 5 is specified. For this solution the quantity u(t, x)/x3 keeps a constant −3/2 value along the trajectory t = kx2 (k = const > 0). For all six cases c1 = t0 u0 , but the expressions for the constant c2 are lengthy and are not given here. We add the following initial condition to integro-differential equation (5.19):
and
The result of an analysis of the invariant H-solutions of system (B) with condition (3.2) is presented in the following theorem. Theorem 6. The set of non-trivial invariant longitudinal displacements of the section of the rod, defined by system (B) with condition (3.2), apart from transformations (4.1)’ and (4.3)’ is exhausted by the displacements specified by formulae (5.2) – (5.6) and (5.17) and (5.19). / 1/2) is presented in the following theorem. The result of an analysis of the invariant H-solutions of system (A) with condition (3.3) (␥ = (A),apart from transformations (4.1) and (4.4), with condition (3.3) Theorem 7. The unique non-trivial invariant H-solution of system (␥ = / 1/2), is a solution invariant with respect to the subgroup Y7 + ␣Y4 , where ␣ is an arbitrary constant. It has the form
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(5.20) Like solution (5.7) it is established that
(5.21) while the function P() is the solution of the integro-differential equation
(5.22) where 0 > 0 and c1 and c2 are arbitrary constants. The solution u = x2␥−1 U() with the function U(), given by formulae (5.17) and (5.18), describes longitudinal oscillations of a viscoelastic rod, the elastic properties of which are given by the relation
while on its end x = x0 > 0 at the initial instant of time t = t0 > 0 one of the six sets of conditions is specified: (5.12) and 1 – 5. For this solution 1/2−␥ the quantity x1−2␥ u(t, x) retains a constant value along the trajectory t = kx2 (k = const > 0). For all six cases c1 = t0 u0 . The initial condition for integro-differential equation (5.22), has the form
and
The expressions for the quantity c2 in these cases are not given here in view of their length. The result of an analysis of the invariant H-solutions of system (B) with condition (3.3) (␥ = / 1/2) is presented in the following theorem. Theorem 8. The set of non-trivial invariant longitudinal displacements of the section of the rod, defined by system (B) with condition (3.3) (␥ = / 1/2), apart from transformations (4.1)’ and (4.4)’ is exhausted by the displacements specified by formulae (5.2) – (5.6) and (5.20) – (5.22). With condition (3.3) (␥ = 1/2) the stress depends on the deformation as given by the relation = u3x . An additional operator for system (A) (system (B)) is the operator Y7 (operator Z8 ) with ␥ = 1/2, while the additional formula for producing the solutions is formula (4.4) (formula (4.4)’). However, by comparison with case (3.3) (␥ = / 1/2) the optimum system of subgroups of the main group of both system (A) and system (B) is changed. An analysis of the invariant H-solutions of system (A) with = u3x is presented in the following theorem. Theorem 9. The unique non-trivial invariant H-solution of system (A) with = u3x , apart from transformations (4.1) and (4.4), is the solution invariant with respect to the subgroup Y7 + ␣Y3 (␥ = 1/2), where ␣ is an arbitrary constant. It has the form (5.23) Like solution (5.7) we have
(5.24) while the function P() is the solution of integro-differential equation
(5.25) where 0 > 0, ␣, c1 and c2 are arbitrary constants. The solution defined by the first formula of (5.23), with function U(), specified by formulae (5.24) and (5.25), describes longitudinal oscillations of a viscoelastic rod, whose elastic properties are defined by the relation = u3x , while on its ends x = x0 > 0 at the initial instant of time t = t0 > 0 one of the six sets of conditions (5.12), 1 – 5 are given. For this solution the quantity u(t, x) – ␣ln x retains a constant value Please cite this article in press as: Chirkunov YuA. Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin’s model. J Appl Math Mech (2016), http://dx.doi.org/10.1016/j.jappmathmech.2016.03.012
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along the trajectories t = kx2 (k = const > 0). In all six cases c1 = u0 − condition
␣ 2
ln t0 . We must add to integro-differential equation (5.25) the initial
and
Expressions for the quantity c2 in these cases are not given here in view of their length. An analysis of the invariant H-solutions of system (B) for = u3x is presented in the following theorem. Theorem 10. The set of non-trivial invariant longitudinal displacements of the section of the rod, defined by system (B) with = u3x , apart from transformations (4.1)’ and (4.4)’ is covered by the variables specified by formulae (5.2) – (5.6), (5.23) – (5.25). Acknowledgements This research was supported financially by the Ministry of Education and Sconce of the Russian Federation (2014/138, No. 435), and the Program for the Support of the Leading Scientific Schools (NSh-2133.2014.1) and the Russian Foundation for Basic Research (12-01-00648). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Greenberg JM, MacCamy RC, Mizel VJ. On the existence, uniqueness, and stability of solutions of the equation ␦’ (ux )uxx + uxtx = 0 utt . J. Math. Mech 1968;17(7):707–28. Greenberg JM. On the existence, uniquencess, and stablity of solutions of the equation 0 ¯ x ) ¯ xx + ¯ xxt . J. Math. Analys. Appl 1969;(25):575–91. ¯ tt = E( Rzhanitsyn AR. Creep Theory. Moscow: Izd Lit Stroitel; 1968. Germain P. Cours de mecanique des milieu continus. T. 1. Toeorie generale. Paris: Masson; 1973. p. 350. Olver PJ. Applications of Lie Groups to Differential Equations. N.Y.: Springer; 1986. p. 570. Chirkunov YuA. On group properties and conservation laws for second-order quasi-linear differential equations. J Appl Mech Tech Phys 2009;50(3):408–13. Chirkunov YuA. Method of A-operators and conservation laws for the equations of gas dynamics. J Appl Mech Tech Phys 2009;50(2):213–9. Chirkunov YuA. Generalized equivalence transformations and group classification of systems of differential equations. J Appl Mech Tech Phys 2012;53(2):147–55. Chirkunov YuA, Khabirov SV. Elements of Symmetrical Analysis of the Differential Equations of Continuum Mechanics. Novosibirsk: Izd NGTU; 2012. Ovsyannikov LV. Group Analysis of Differential Equations. Moscow: Nauka; 1978.
Translated by R.C.G.
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