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The development and deceleration of the flow of an elastoplastic medium in a cylindrical tube*
Journal of Applied Mathematics and Mechanics 77 (2013) 566–572
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The development and deceleration of the flow of an elastoplastic medium in a cylindrical tube*夽 A.A. Burenin ∗ , L.V. Kovtanyuk Vladivostok, Russia
a r t i c l e
i n f o
Article history: Received 14 February 2013
a b s t r a c t An exact solution of the quasistatic problem of elastoplastic theory of the development of the flow of an incompressible medium in a cylindrical tube of circular cross section due to an increase in the pressure drop with time, and on the subsequent flow when there is a constant pressure and a deceleration due to its slow reduction. The conditions for the occurrence and regularity of the advancement of the elastoplastic boundaries for different types of loading are indicated. © 2014 Elsevier Ltd. All rights reserved.
A number of exact solutions of the problems of antiplane flow1–4 have been obtained using the rigid-plastic Shvedov–Bingham model, and fairly universal methods of calculating viscoplastic flows have been developed.5–7 Mathematical simulation of the flows is considerably more complicated if we reject the assumption that the medium comprising rigid cores, or in stagnation zones, is undeformable. Deformations in these regions are mainly reversible, and the boundary value problems must necessarily be formulated in terms of displacements, while in the flow regions the problem is solved in terms of the rates of displacement. At the boundaries of the regions one must require that the conditions of continuity of the displacements are satisfied, since the equality of the velocities and the components of the stresses is insufficient and may lead to erroneous solutions.8 A calculation of the components of these stresses in flow regions, as is well known,9 is not a simple problem. In such regions irreversible deformations are necessarily large, which requires considerations using the large elastoplastic deformation model. Many such models have been proposed, beginning with the first geometrically non-contradictory model.10 We give only some of the publications by Russian authors in Refs 11–16 . We will use the mathematical model proposed in Ref. 17 and described earlier in Ref. 18 . This model is specific, contains no new constants of the medium and corresponds to the classical requirements of the elastoplastic model: irreversible deformations are not changed when unloading occurs, reversible deformations completely specify the stresses in the medium, and the unloading state is independent of the type of unloading in stress space. These requirements are not obligatory for the model of large elastoplastic deformations, and sometimes they are also not experimentally justified, but the hypothetical formulation of these conditions leads to the simplest version of the model, which enables us, using examples of the solution of the simplest model problems, to discuss the correctness of their formulation and, in individual cases obtain exact solutions. The availability of these solutions19–21 for basically non-linear theory is an extremely important fact. In this paper we construct an exact solution of the classical problem of the flow of an elastoplastic medium in a cylindrical tube under variable pressure-drop conditions. A similar problem of the flow of a plug was considered in Ref. 19 , when the medium occupies a finite volume in the tube and is loaded with a pressure on one of the free boundaries. The finite displacement of the plug is a consequence of the viscoplastic flow of the medium in the boundary layer.
1. Initial relations of the deformation model. We will assume that the state parameters of an isothermally deformable medium are two symmetrical kinematic tensors with components eij and pij . Using the well-known approach15 and following the formalism of non-equilibrium thermodynamics, for these tensors we will postulate the equations of their variation (transposition) in the form
夽 Prikl. Mat. Mekh., Vol. 77, No. 5, pp. 788-798, 2013. ∗ Corresponding author. E-mail address: [email protected] (A.A. Burenin). 0021-8928/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.12.012
A.A. Burenin, L.V. Kovtanyuk / Journal of Applied Mathematics and Mechanics 77 (2013) 566–572
567
(1.1) In these equations we have used a rectangular system of Euler Cartesian coordinates xi ,. and ui and vi are the components of the p displacement and velocity vectors of points of the medium. When the source εij in the transposition equation of the tensor (pij ) is equal to zero, the components of the tensor (pij ) are changed in the same way as when the system of coordinates is rotated, or, which is the same thing, when the medium moves without deformation. p We identify this tensor with the irreversible deformation tensor, and we will connect the unloading process with the equality εij = 0. Then the tensor (eij ) could be assumed to be the characteristic of the reversible deformation, while relations (1.1) are the definitions of the reversible and irreversible deformations. However, the unknown components zij and the objective derivative operator D/Dr remain in these relations. We obtain a model that is geometrically and kinematically noncontradictory17,18 for the following necessary definition of the objective derivative with respect to time, written for the arbitrary tensor (nij )
(1.2) Note that when the non-linear component (zij ) of the rotation tensor (rij ) is equal to zero, derivative (1.2) becomes the Jaumann derivative. Relations (1.1) and (1.2) dictate the following separation of the total Almansi strains on the reversible and irreversible components
(1.3) The tensor with components eij specifies the irreversible deformation tensor with components sij , but is not identical with the latter. The introduction into consideration of a tensor with components eij is not only for convenience in writing its transposition equation (1.1), but is connected with the possibility of writing Murnaghan’s formula as a corollary of the law of conservation of energy. If we assume that, in the case of an elastoplastic medium, the thermodynamic potential (the free energy distribution density) is a function solely of the reversible deformations, then, from the law of conservation of energy we can obtain18
(1.4) Relations (1.4) are written for the case of an incompressible medium, considered below, P and P1 are additional hydrostatic pressures, W = W(eij ) is the elastic potential, which, assuming the medium to be isotropic, can be specified in the form22
(1.5) The parameter is usually identified with the shear modulus and b and are elastic constants of higher order. It follows from relations (1.4) and (1.5) that, over the whole region of deformation, the stresses in the medium are determined by the level and distribution of the reversible deformations. We will assume that the irreversible deformations are stored in the medium in conditions corresponding to the stresses of the load surface, while their rate of growth is specified by the associated law of plastic flow, i.e., (1.6) A new constant of the material k – the yield point, is introduced by relations (1.6). We will later use the Tresca plasticity condition as a function of the load, generalized to the case when the viscous properties of the medium are taken into account in its plastic flow:23 (1.7) Here i are the components of the principal stresses, viscosity coefficient.
p εk
are the components of the main rates of plastic deformation, and is the
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2. Elastic deformation. Suppose an incompressible elastoviscoplastic material, the deformation properties of which are specified above, fills a cylindrical tube of radius R with undeformable walls. Consider the deformation of the material and its motion along the tube when there is a pressure drop that increases with time. The solution of this boundary value problem of the theory of elastoviscoplastic deformations in a cylindrical system of coordinates r, , z will be sought in the class of functions
According to relations (1.3), only two components of the deformation tensor are non-zero in this case:
(2.1)
We will assume that the deformation begins from the free state of the material and is initially reversible dij = s, pij = 0 . Following Murnaghan’s formula (1.4), we obtain for the elastic potential (1.5)
(2.2) Here s(z, r, t) is a new unknown function of the additional hydrostatic pressure. The fact that rr and are independent of r will be established below. In relations (2.2) we have not included terms with third and higher powers of u,r. Hence, the reversible deformations are assumed to be small and we only take into account leading non-linear terms in the stress – reversible deformation relations. This limitation is not essential, but it enables us to obtain exact solutions of the sequence of boundary value problems of the development and retardation of the flow in visible form. For materials that are most characteristic in exhibiting elastoplastic properties (metals), this limitation is natural. Further, we will neglect inertia forces, assuming that the deformation and flow processes are fairly slow. It will then follow from the equilibrium equations that, of the diagonal components of the stress tensor, only zz depends on r, while rr = = −p (z, t) is independent of r. Integration of the equilibrium equations leads to the relations (2.3) The unknown integration functions c(t), c1 (t) and p0 (t) must be determined from the boundary conditions. Note that c1 (t) = 0, since the stress rz is necessarily finite when r = 0, and p0 (t) must be assumed to be a known (specified) function of the control pressure in the tube cross section z = 0. We will relate the deformation and motion of the material along the tube with the action of the pressure gradient (2.4) We will assume that as long as (2.5) where 0 is the specified dry friction constant ( 0 < k), the strict adhesion condition u(R,t) = 0 is satisfied. For these loading conditions, we obtain the final solution (2.6) This solution remains admissible as the function (t) increases up to the instant of time t = t*, so long as strict inequality (2.5) is satisfied. To be specific, we will further assume that (t) is a linear function of time (t) = ␣t (␣ = const). In this case t* = 20 /(␣R). Beginning from the instant of time t = t*, slippage begins on the wall r = R. The no-slip condition is now not satisfied and is replaced by the boundary condition (2.7) where is the viscous friction constant. The change in the mode of deformation, connected with the start of slippage, does not change relation (2.6) for the stresses. The following relations hold for the displacement and the velocity
(2.8) where x(t) is an unknown function, determined from the boundary condition. Using the law of viscous friction (2.7), we finally obtain
(2.9) Solution (2.9) holds in the time interval from t* to the following instant of time t = t0 , beginning from which a region of plastic flow develops from the wall r = R. The condition for it to occur (1.7), with the assumptions made, takes the form (2.10)
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According to formula (1.1), the elastic deformations are expressed in terms of the displacement field (2.9) by the relations
(2.11) 3. The developing plastic flow and the flow at constant pressure drop. At any actual instant of time t following t0 , a flow region m(t) ≤ r ≤ R is present in the medium. The moving surface r = m(t) isolates a region of flow from the elastic kernel r ≤ m(t), where only reversible (elastic) deformations are present in the medium. In the region of reversible deformation, as previously, relations (2.6) are satisfied for the stresses and (2.8) for the displacements and velocities. In the flow region m(t) ≤ r ≤ R, according to relations (1.4) when pij = / 0, (1.5) and (2.11), we have
(3.1) A consequence of the integration of the equilibrium equations, as previously, is the fact that the function p1 is independent of r, and we also have the expressions (3.2) The stress continuity conditions on the elastoplastic boundary r = m(t) compels us to assume that
Plastic flow condition (1.7) in the case considered can be rewritten in the form (3.3) Following the associated plastic flow law (1.6), we obtain from condition (3.3)
(3.4) A comparison of relations (3.2) and (3.4), taking condition (2.4) into account, enables us to calculate the rate of plastic deformations (3.5) The position of the elastoplastic boundary r = m(t) is specified by the condition that the rate of plastic deformations (3.5) is equal to zero on this boundary: (3.6) The velocity distribution in the flow region follows from equality (3.5), taking boundary condition (2.7) into account
(3.7) Using the condition of continuity of velocities (2.8) and (3.7) on the elastoplastic boundary, we obtain in the region of the elastic kernel r ≤ m(t)
(3.8) As already noted, in elastoplasticity problems it is necessary to construct a displacement field that is continuous on the elastoplastic boundary. To obtain the displacements in the flow region it is necessary to determine the component of the plastic deformation prz . Integrating Eq. (3.5) with the condition that the plastic deformation must equal zero on the boundary r = m(t), we obtain
(3.9) From the known elastic deformation (3.2) and the plastic deformation (3.9), using relations (1.1) and (2.1), we determine the total deformations and, consequently, the displacements in the flow region
(3.10) Here we have used the condition of continuity of the displacement when r = m(t).
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Hence, the solution of the problem of the development of viscoplastic flow as the pressure drop increases can be taken as completed. We emphasise that the choice of the simplest relation (t) = ␣t is due solely to the specific definition of the problem and the fact that we have obtained the simplest final relationships, which solve the problem. The calculations can be repeated for any other monotonically increasing function (t). Suppose, beginning from an instant of time t = t1 , the function (t) remains constant: (t) = ␣t1 . Then, by relations (3.6), the flow region is not increased while the plastic deformations grow within the layer m(t1 ) = m1 ≤ r ≤ R. The process of the buildup of plastic deformations and the deformed state in the moving elastic kernel r ≤ m1 obey relations which follow directly from the solution obtained above: in the elastic kernel
(3.11) in the flow region (3.12)
In (3.12) one must use the relation for the reversible deformations from relations (3.2). 4. Flow when there is a reduction in the pressure drop. Pressure relief. Suppose, beginning from a certain instant of time t = t2 > t1 , the function (t) decreases, for example, as follows: (4.1) At any instant of time t, following t2 , three different deformed regions will exist in the medium: the region of the elastic kernel r ≤ m1 , the region m1 ≤ r ≤ l(t), where there are irreversible deformations, but they do not build up, and the region where the flow continues l(t) ≤ r ≤ R. The cylindrical surface r = l(t) will be a new elastoplastic boundary, moving from the surface r = m1 to the outer boundary of the medium r = R. In other words, the radius of the elastic kernel, which moves as a deformed whole, increases due to the connection to it of the region m1 ≤ r ≤ l(t), where, in addition to reversible deformations, there are also non-variable irreversible deformations. Note that the inverse deformation tensor remains unchanged, whereas its components may vary. In the case considered the quantity prz is unchanged, while prr and pzz vary in this region. In the elastic region, like the previous relations for any instant of time t > t2 we obtain for the displacements and velocities
(4.2) Here w(t) is an unknown function, determined from the boundary condition. At each point of the region m1 ≤ r ≤ R the flow continues up to the instant of time when it reaches the moving surface l(t). After this, in the whole region m1 ≤ r ≤ l(t) the rate of plastic deformation p εrz = 0, and, consequently, the quantity prz remains constant. This leads to the fact that, in the region indicated, relations (4.2) hold. In the p p region l(t) ≤ r ≤ R, where the flow continues, similar to the derivation of relation (3.5), we obtain εrz . The requirement εrz (l(t)) = 0 enables us to write the law of motion of the boundary r = l(t): (4.3) As previously, using kinematic relations (1.1) and (3.5) and also condition (2.7), we determine the velocity in the region of continuous flow
(4.4) The condition for the velocities on the boundary of the flow region r = l(t) to be equal enables us to find the unknown function w(t) in relations (4.2) and to obtain that the function v = F(l, t) specifies the time dependence of the velocity of motion of the kernel. To determine the displacements in the flow region it is necessary to calculate the component prz of the irreversible deformations. Integrating equality (3.5), taking (4.2) into account and using the initial condition at the instant when the flow regime changes when t = t2 , we obtain an expression for prz which differs from that derived at the end of Section 3 in having the term ˛rt1 ˇ(t − t2 )2 /(4) on the right-hand side. Hence, when the law of motion of the elastoplastic boundary (4.3) is known, we can determine the irreversible deformations in the region of the kernel m1 ≤ r ≤ l(t)
(4.5)
A.A. Burenin, L.V. Kovtanyuk / Journal of Applied Mathematics and Mechanics 77 (2013) 566–572
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From the known velocities of points of the medium, and the elastic and plastic deformations, we obtain the displacements, taking their equality on the boundaries m1 and l(t) into account: in the plastic-flow region l(t) ≤ r ≤ R
(4.6) where (4.7) in the region m1 ≤ r ≤ l(t), adjoining the kernel, (4.8) in the elastic region r ≤ m1 (4.9) The elastoplastic boundary r = l(t) in its motion in the medium, at the instant of time
reaches the boundary r = R. Consequently, at the instant t = t3 the viscoelastic flow degenerates into slippage of the kernel along the walls of the tube when there is viscous friction. The displacements and velocities at the instant t = t3 are determined by relations (4.6) and (4.8) when m1 ≤ r ≤ R and (4.6) and (4.9) when r ≤ m1 respectively, if we put l = R in them. When the function (t) decreases further (t > t3 ) in accordance with (4.1), the velocity distribution over the whole medium conforms to relation (4.4) when g(r, t) ≡ 0. The displacements are also determined by relations (4.6), (4.8) and (4.9) when l = R. At the instant of time
the velocity (4.4) becomes equal to zero on the surface r = R. From this instant, up to the final instant of unloading tk = −1 + t2 when r = R the no-slip condition (R) = 0 is satisfied. The velocity in the intermediate period t4 ≤ t ≤ tk is determined from the formula (4.10) In the same time interval, the displacements are determined by the relations
(4.11) when 1 = 1, rˆ = r in the region m1 ≤ r ≤ R and when 1 = 0, rˆ = m1 in the elastic deformation region. At the final instant of pressure relief t = tk the first term in the expression for the function h3 (r, t) becomes equal to zero. Consequently, from this instant of time the motion of the medium stops and displacements then remain constant. Note that, at this instant, the velocity of motion of the points of the medium in the kernel changes abruptly. Up to the instant t = tk the velocity distribution is given by relation (4.10) (t < tk ), but from this instant (t > tk ) the velocity becomes zero. The same occurred when the loading modes with an increasing pressure drop were replaced by a constant pressure drop and a constant pressure drop was replaced by a reducing pressure drop. Consequently, a discontinuity in the derivative with respect to time of the function (t) = ∂p/∂z leads to a sudden change in the velocity distribution. The essential non-stationary nature of this change obviously requires the force of inertia to be taken into account as well as the related wave velocity redistribution, which is outside the framework of the quasistatic approach used here. As an illustration of the features of the flow and the deformation in the kernel when the elastic properties of the medium are taken into account, Fig. 1 shows graphs of the distribution of the dimensionless velocity ␥ = v/(R) as a function of the dimensionless spatial coordinate = r/R at different instants of dimensionless time = ␣Rt/, which follow from the solution obtained of the succession of problems of the development and retardation of the flow. The following values of the constants were taken
Note that the velocity of the medium increases as the pressure drop increases. At a constant pressure drop, the elastic kernel moves as a deformed whole more rapidly than points of the medium in the flow region. When the pressure drop is reduced, the motion of the elastic kernel, due to the elastic effect, slows down more rapidly compared with the motion in the region of viscoplastic flow. An even
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Fig. 1.
clearer elastic consequence occurs when the flow stops, while in the case when slippage ceases all points of the medium will move in the opposite direction and the elastic strains fall to zero when the pressure drop is completely removed ( = 0). The analytical solution obtained may be useful in calculations of technological metal drawing operations (the numerical values of the constants given above are for steel), the transportation of heavy hydrocarbons etc., and serve as a basis for testing algorithms and numerical calculation programs. Note, finally, that we have used here the simplest relations for the friction, both dry and viscous. Corresponding calculations can be carried out for more complex friction laws. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
Myasnikov VP. Some exact solutions for the rectilinear motions of a viscoplastic medium. Zh Prikl Mekh Tekh Fiz 1961;2:54–60. Bykovtsev GI, Chernyshov AD. Viscoplastic flow in non-circular cylinders when there is a pressure drop. Zh Prikl Mekh Tekh Fiz 1964;4:94–6. Ogibalov PM, Mirzadzhanzade AKh. Unsteady Motions of Viscoplastic Media. Moscow: Izd MGU; 1970. Georgiyevskii DV. Rigid zones in statically defined and undefined problems of viscoplastic flow. In: Problems of the Mechanics of Deformable Solids and Rocks. Collected Papers celebrating the 75th birth day of Ye. I. Shemyakin. 2006. Rezunov AV, Chernyshev AD. The problem of pure shear of a viscoplastic material between two cylindrical surfaces. In: Mechanics of a Deformable Solid Kuibyshev Izd Kuibyshev Univ: 1975;1:32–6. Duvant G, Lions J-L. Les Inequalitions en Mécanique et en Physique. Paris: Dunod; 1972. Mosolov PP, Myasnikov VP. Mechanics of Rigid Plastic Media. Moscow: Nauka; 1981. Ivlev DD. From the history of discussions in mechanics. Three discussions Theoretical and Applied Mechanics International Collection Minsk; BNTU; 2012;27:5–10. Ivlev DD. Determination of displacements in elastoplastic problems of the theory of ideal plasticity. Advances in the Mechanics of Deformable Media. Celebrating the 100th Anniversary of the Birth of V. G. Galerkin. Moscow: Nauka: 1975. Lee EH. Elastic-plastic deformation at finite strains. Trans ASME Ser E J Appl Mech 1969;36(1):1–6. Magomedov OB, Pobedrya BYe. Some problems of viscoelastoplastic flow. Elasticity and Non-Elasticity. Moscow; Izd MGU: 1975;4:152–69. Kondaurov VI. The equations of an elastoviscoplastic medium with finite strains. Zh Prikl Mekh Tekh Fiz 1982;4:133–9. Levitas VI. Large Elastoplastic Deformations of Materials at High Pressure. Naukova Dumka: Kiev; 1987. Myasnikov VP. The equations of motion of elastoplastic materials for large strains. Vestnik DVO Ross Akad Nauk 1996;4:8–13. Chernyshov AD. The consititutive relations for an elastoplastic solid for finite strains. Izv Ross Akad Nauk MTT 2000;1:120–8. Rogovoi AA. Consititutive relations for finite elastic-inelastic strains. Zh Prikl Mekh Tekh Fiz 2005;46(5):138–49. Burenin AA, Bykovtsev GI, Kovtanyuk LV. A simple model for an elastoplastic medium for finite strains. Dokl Ross Akad Nauk 1996;347(2):199–201. Burenin AA, Kovtanyuk LV. Elastic Effects in Intensive Irreversible Strain. Vladivostok: Izd DVGTU; 2011. Kovtanyuk LV. The extrusion of an elastoviscoplastic material through a rigid circular cylindrical matrix. Dokl Ross Akad Nauk 2005;400(6):764–7. Burenin AA, Kovtanyuk LV, Mazelis AL. The pressing of an elastoviscoplastic material between rigid coaxial cylindrical surfaces. J Appl Math Mech 2006;70(3):437–45. Burenin AA, Ustinova AS. The development and drag of a helical viscoplastic flow with a calculation of the elastic response after flow and unloading stops. In Advances in Continuum Mechanics Collected Papers on the 70th Birthday of V A Levin Vladivostok: Dal’Nauka; 2009:91–102. Lurie AI. Non linear Theory of Elasticity. Amsterdam: North-Holland; 1980. Znamenskii VA, Ivlev DD. Equations of a viscoplastic solid for piecewise-linear potentials. Izd Akad Nauk SSSR OTN Mekhanika i Mashinostroyeniye 1963;6:12–6.
Translated by R.C.G.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The development and deceleration of the flow of an elastoplastic medium in a cylindrical tube*夽 A.A. Burenin ∗ , L.V. Kovtanyuk Vladivostok, Russia
a r t i c l e
i n f o
Article history: Received 14 February 2013
a b s t r a c t An exact solution of the quasistatic problem of elastoplastic theory of the development of the flow of an incompressible medium in a cylindrical tube of circular cross section due to an increase in the pressure drop with time, and on the subsequent flow when there is a constant pressure and a deceleration due to its slow reduction. The conditions for the occurrence and regularity of the advancement of the elastoplastic boundaries for different types of loading are indicated. © 2014 Elsevier Ltd. All rights reserved.
A number of exact solutions of the problems of antiplane flow1–4 have been obtained using the rigid-plastic Shvedov–Bingham model, and fairly universal methods of calculating viscoplastic flows have been developed.5–7 Mathematical simulation of the flows is considerably more complicated if we reject the assumption that the medium comprising rigid cores, or in stagnation zones, is undeformable. Deformations in these regions are mainly reversible, and the boundary value problems must necessarily be formulated in terms of displacements, while in the flow regions the problem is solved in terms of the rates of displacement. At the boundaries of the regions one must require that the conditions of continuity of the displacements are satisfied, since the equality of the velocities and the components of the stresses is insufficient and may lead to erroneous solutions.8 A calculation of the components of these stresses in flow regions, as is well known,9 is not a simple problem. In such regions irreversible deformations are necessarily large, which requires considerations using the large elastoplastic deformation model. Many such models have been proposed, beginning with the first geometrically non-contradictory model.10 We give only some of the publications by Russian authors in Refs 11–16 . We will use the mathematical model proposed in Ref. 17 and described earlier in Ref. 18 . This model is specific, contains no new constants of the medium and corresponds to the classical requirements of the elastoplastic model: irreversible deformations are not changed when unloading occurs, reversible deformations completely specify the stresses in the medium, and the unloading state is independent of the type of unloading in stress space. These requirements are not obligatory for the model of large elastoplastic deformations, and sometimes they are also not experimentally justified, but the hypothetical formulation of these conditions leads to the simplest version of the model, which enables us, using examples of the solution of the simplest model problems, to discuss the correctness of their formulation and, in individual cases obtain exact solutions. The availability of these solutions19–21 for basically non-linear theory is an extremely important fact. In this paper we construct an exact solution of the classical problem of the flow of an elastoplastic medium in a cylindrical tube under variable pressure-drop conditions. A similar problem of the flow of a plug was considered in Ref. 19 , when the medium occupies a finite volume in the tube and is loaded with a pressure on one of the free boundaries. The finite displacement of the plug is a consequence of the viscoplastic flow of the medium in the boundary layer.
1. Initial relations of the deformation model. We will assume that the state parameters of an isothermally deformable medium are two symmetrical kinematic tensors with components eij and pij . Using the well-known approach15 and following the formalism of non-equilibrium thermodynamics, for these tensors we will postulate the equations of their variation (transposition) in the form
夽 Prikl. Mat. Mekh., Vol. 77, No. 5, pp. 788-798, 2013. ∗ Corresponding author. E-mail address: [email protected] (A.A. Burenin). 0021-8928/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.12.012
A.A. Burenin, L.V. Kovtanyuk / Journal of Applied Mathematics and Mechanics 77 (2013) 566–572
567
(1.1) In these equations we have used a rectangular system of Euler Cartesian coordinates xi ,. and ui and vi are the components of the p displacement and velocity vectors of points of the medium. When the source εij in the transposition equation of the tensor (pij ) is equal to zero, the components of the tensor (pij ) are changed in the same way as when the system of coordinates is rotated, or, which is the same thing, when the medium moves without deformation. p We identify this tensor with the irreversible deformation tensor, and we will connect the unloading process with the equality εij = 0. Then the tensor (eij ) could be assumed to be the characteristic of the reversible deformation, while relations (1.1) are the definitions of the reversible and irreversible deformations. However, the unknown components zij and the objective derivative operator D/Dr remain in these relations. We obtain a model that is geometrically and kinematically noncontradictory17,18 for the following necessary definition of the objective derivative with respect to time, written for the arbitrary tensor (nij )
(1.2) Note that when the non-linear component (zij ) of the rotation tensor (rij ) is equal to zero, derivative (1.2) becomes the Jaumann derivative. Relations (1.1) and (1.2) dictate the following separation of the total Almansi strains on the reversible and irreversible components
(1.3) The tensor with components eij specifies the irreversible deformation tensor with components sij , but is not identical with the latter. The introduction into consideration of a tensor with components eij is not only for convenience in writing its transposition equation (1.1), but is connected with the possibility of writing Murnaghan’s formula as a corollary of the law of conservation of energy. If we assume that, in the case of an elastoplastic medium, the thermodynamic potential (the free energy distribution density) is a function solely of the reversible deformations, then, from the law of conservation of energy we can obtain18
(1.4) Relations (1.4) are written for the case of an incompressible medium, considered below, P and P1 are additional hydrostatic pressures, W = W(eij ) is the elastic potential, which, assuming the medium to be isotropic, can be specified in the form22
(1.5) The parameter is usually identified with the shear modulus and b and are elastic constants of higher order. It follows from relations (1.4) and (1.5) that, over the whole region of deformation, the stresses in the medium are determined by the level and distribution of the reversible deformations. We will assume that the irreversible deformations are stored in the medium in conditions corresponding to the stresses of the load surface, while their rate of growth is specified by the associated law of plastic flow, i.e., (1.6) A new constant of the material k – the yield point, is introduced by relations (1.6). We will later use the Tresca plasticity condition as a function of the load, generalized to the case when the viscous properties of the medium are taken into account in its plastic flow:23 (1.7) Here i are the components of the principal stresses, viscosity coefficient.
p εk
are the components of the main rates of plastic deformation, and is the
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2. Elastic deformation. Suppose an incompressible elastoviscoplastic material, the deformation properties of which are specified above, fills a cylindrical tube of radius R with undeformable walls. Consider the deformation of the material and its motion along the tube when there is a pressure drop that increases with time. The solution of this boundary value problem of the theory of elastoviscoplastic deformations in a cylindrical system of coordinates r, , z will be sought in the class of functions
According to relations (1.3), only two components of the deformation tensor are non-zero in this case:
(2.1)
We will assume that the deformation begins from the free state of the material and is initially reversible dij = s, pij = 0 . Following Murnaghan’s formula (1.4), we obtain for the elastic potential (1.5)
(2.2) Here s(z, r, t) is a new unknown function of the additional hydrostatic pressure. The fact that rr and are independent of r will be established below. In relations (2.2) we have not included terms with third and higher powers of u,r. Hence, the reversible deformations are assumed to be small and we only take into account leading non-linear terms in the stress – reversible deformation relations. This limitation is not essential, but it enables us to obtain exact solutions of the sequence of boundary value problems of the development and retardation of the flow in visible form. For materials that are most characteristic in exhibiting elastoplastic properties (metals), this limitation is natural. Further, we will neglect inertia forces, assuming that the deformation and flow processes are fairly slow. It will then follow from the equilibrium equations that, of the diagonal components of the stress tensor, only zz depends on r, while rr = = −p (z, t) is independent of r. Integration of the equilibrium equations leads to the relations (2.3) The unknown integration functions c(t), c1 (t) and p0 (t) must be determined from the boundary conditions. Note that c1 (t) = 0, since the stress rz is necessarily finite when r = 0, and p0 (t) must be assumed to be a known (specified) function of the control pressure in the tube cross section z = 0. We will relate the deformation and motion of the material along the tube with the action of the pressure gradient (2.4) We will assume that as long as (2.5) where 0 is the specified dry friction constant ( 0 < k), the strict adhesion condition u(R,t) = 0 is satisfied. For these loading conditions, we obtain the final solution (2.6) This solution remains admissible as the function (t) increases up to the instant of time t = t*, so long as strict inequality (2.5) is satisfied. To be specific, we will further assume that (t) is a linear function of time (t) = ␣t (␣ = const). In this case t* = 20 /(␣R). Beginning from the instant of time t = t*, slippage begins on the wall r = R. The no-slip condition is now not satisfied and is replaced by the boundary condition (2.7) where is the viscous friction constant. The change in the mode of deformation, connected with the start of slippage, does not change relation (2.6) for the stresses. The following relations hold for the displacement and the velocity
(2.8) where x(t) is an unknown function, determined from the boundary condition. Using the law of viscous friction (2.7), we finally obtain
(2.9) Solution (2.9) holds in the time interval from t* to the following instant of time t = t0 , beginning from which a region of plastic flow develops from the wall r = R. The condition for it to occur (1.7), with the assumptions made, takes the form (2.10)
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According to formula (1.1), the elastic deformations are expressed in terms of the displacement field (2.9) by the relations
(2.11) 3. The developing plastic flow and the flow at constant pressure drop. At any actual instant of time t following t0 , a flow region m(t) ≤ r ≤ R is present in the medium. The moving surface r = m(t) isolates a region of flow from the elastic kernel r ≤ m(t), where only reversible (elastic) deformations are present in the medium. In the region of reversible deformation, as previously, relations (2.6) are satisfied for the stresses and (2.8) for the displacements and velocities. In the flow region m(t) ≤ r ≤ R, according to relations (1.4) when pij = / 0, (1.5) and (2.11), we have
(3.1) A consequence of the integration of the equilibrium equations, as previously, is the fact that the function p1 is independent of r, and we also have the expressions (3.2) The stress continuity conditions on the elastoplastic boundary r = m(t) compels us to assume that
Plastic flow condition (1.7) in the case considered can be rewritten in the form (3.3) Following the associated plastic flow law (1.6), we obtain from condition (3.3)
(3.4) A comparison of relations (3.2) and (3.4), taking condition (2.4) into account, enables us to calculate the rate of plastic deformations (3.5) The position of the elastoplastic boundary r = m(t) is specified by the condition that the rate of plastic deformations (3.5) is equal to zero on this boundary: (3.6) The velocity distribution in the flow region follows from equality (3.5), taking boundary condition (2.7) into account
(3.7) Using the condition of continuity of velocities (2.8) and (3.7) on the elastoplastic boundary, we obtain in the region of the elastic kernel r ≤ m(t)
(3.8) As already noted, in elastoplasticity problems it is necessary to construct a displacement field that is continuous on the elastoplastic boundary. To obtain the displacements in the flow region it is necessary to determine the component of the plastic deformation prz . Integrating Eq. (3.5) with the condition that the plastic deformation must equal zero on the boundary r = m(t), we obtain
(3.9) From the known elastic deformation (3.2) and the plastic deformation (3.9), using relations (1.1) and (2.1), we determine the total deformations and, consequently, the displacements in the flow region
(3.10) Here we have used the condition of continuity of the displacement when r = m(t).
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Hence, the solution of the problem of the development of viscoplastic flow as the pressure drop increases can be taken as completed. We emphasise that the choice of the simplest relation (t) = ␣t is due solely to the specific definition of the problem and the fact that we have obtained the simplest final relationships, which solve the problem. The calculations can be repeated for any other monotonically increasing function (t). Suppose, beginning from an instant of time t = t1 , the function (t) remains constant: (t) = ␣t1 . Then, by relations (3.6), the flow region is not increased while the plastic deformations grow within the layer m(t1 ) = m1 ≤ r ≤ R. The process of the buildup of plastic deformations and the deformed state in the moving elastic kernel r ≤ m1 obey relations which follow directly from the solution obtained above: in the elastic kernel
(3.11) in the flow region (3.12)
In (3.12) one must use the relation for the reversible deformations from relations (3.2). 4. Flow when there is a reduction in the pressure drop. Pressure relief. Suppose, beginning from a certain instant of time t = t2 > t1 , the function (t) decreases, for example, as follows: (4.1) At any instant of time t, following t2 , three different deformed regions will exist in the medium: the region of the elastic kernel r ≤ m1 , the region m1 ≤ r ≤ l(t), where there are irreversible deformations, but they do not build up, and the region where the flow continues l(t) ≤ r ≤ R. The cylindrical surface r = l(t) will be a new elastoplastic boundary, moving from the surface r = m1 to the outer boundary of the medium r = R. In other words, the radius of the elastic kernel, which moves as a deformed whole, increases due to the connection to it of the region m1 ≤ r ≤ l(t), where, in addition to reversible deformations, there are also non-variable irreversible deformations. Note that the inverse deformation tensor remains unchanged, whereas its components may vary. In the case considered the quantity prz is unchanged, while prr and pzz vary in this region. In the elastic region, like the previous relations for any instant of time t > t2 we obtain for the displacements and velocities
(4.2) Here w(t) is an unknown function, determined from the boundary condition. At each point of the region m1 ≤ r ≤ R the flow continues up to the instant of time when it reaches the moving surface l(t). After this, in the whole region m1 ≤ r ≤ l(t) the rate of plastic deformation p εrz = 0, and, consequently, the quantity prz remains constant. This leads to the fact that, in the region indicated, relations (4.2) hold. In the p p region l(t) ≤ r ≤ R, where the flow continues, similar to the derivation of relation (3.5), we obtain εrz . The requirement εrz (l(t)) = 0 enables us to write the law of motion of the boundary r = l(t): (4.3) As previously, using kinematic relations (1.1) and (3.5) and also condition (2.7), we determine the velocity in the region of continuous flow
(4.4) The condition for the velocities on the boundary of the flow region r = l(t) to be equal enables us to find the unknown function w(t) in relations (4.2) and to obtain that the function v = F(l, t) specifies the time dependence of the velocity of motion of the kernel. To determine the displacements in the flow region it is necessary to calculate the component prz of the irreversible deformations. Integrating equality (3.5), taking (4.2) into account and using the initial condition at the instant when the flow regime changes when t = t2 , we obtain an expression for prz which differs from that derived at the end of Section 3 in having the term ˛rt1 ˇ(t − t2 )2 /(4) on the right-hand side. Hence, when the law of motion of the elastoplastic boundary (4.3) is known, we can determine the irreversible deformations in the region of the kernel m1 ≤ r ≤ l(t)
(4.5)
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From the known velocities of points of the medium, and the elastic and plastic deformations, we obtain the displacements, taking their equality on the boundaries m1 and l(t) into account: in the plastic-flow region l(t) ≤ r ≤ R
(4.6) where (4.7) in the region m1 ≤ r ≤ l(t), adjoining the kernel, (4.8) in the elastic region r ≤ m1 (4.9) The elastoplastic boundary r = l(t) in its motion in the medium, at the instant of time
reaches the boundary r = R. Consequently, at the instant t = t3 the viscoelastic flow degenerates into slippage of the kernel along the walls of the tube when there is viscous friction. The displacements and velocities at the instant t = t3 are determined by relations (4.6) and (4.8) when m1 ≤ r ≤ R and (4.6) and (4.9) when r ≤ m1 respectively, if we put l = R in them. When the function (t) decreases further (t > t3 ) in accordance with (4.1), the velocity distribution over the whole medium conforms to relation (4.4) when g(r, t) ≡ 0. The displacements are also determined by relations (4.6), (4.8) and (4.9) when l = R. At the instant of time
the velocity (4.4) becomes equal to zero on the surface r = R. From this instant, up to the final instant of unloading tk = −1 + t2 when r = R the no-slip condition (R) = 0 is satisfied. The velocity in the intermediate period t4 ≤ t ≤ tk is determined from the formula (4.10) In the same time interval, the displacements are determined by the relations
(4.11) when 1 = 1, rˆ = r in the region m1 ≤ r ≤ R and when 1 = 0, rˆ = m1 in the elastic deformation region. At the final instant of pressure relief t = tk the first term in the expression for the function h3 (r, t) becomes equal to zero. Consequently, from this instant of time the motion of the medium stops and displacements then remain constant. Note that, at this instant, the velocity of motion of the points of the medium in the kernel changes abruptly. Up to the instant t = tk the velocity distribution is given by relation (4.10) (t < tk ), but from this instant (t > tk ) the velocity becomes zero. The same occurred when the loading modes with an increasing pressure drop were replaced by a constant pressure drop and a constant pressure drop was replaced by a reducing pressure drop. Consequently, a discontinuity in the derivative with respect to time of the function (t) = ∂p/∂z leads to a sudden change in the velocity distribution. The essential non-stationary nature of this change obviously requires the force of inertia to be taken into account as well as the related wave velocity redistribution, which is outside the framework of the quasistatic approach used here. As an illustration of the features of the flow and the deformation in the kernel when the elastic properties of the medium are taken into account, Fig. 1 shows graphs of the distribution of the dimensionless velocity ␥ = v/(R) as a function of the dimensionless spatial coordinate = r/R at different instants of dimensionless time = ␣Rt/, which follow from the solution obtained of the succession of problems of the development and retardation of the flow. The following values of the constants were taken
Note that the velocity of the medium increases as the pressure drop increases. At a constant pressure drop, the elastic kernel moves as a deformed whole more rapidly than points of the medium in the flow region. When the pressure drop is reduced, the motion of the elastic kernel, due to the elastic effect, slows down more rapidly compared with the motion in the region of viscoplastic flow. An even
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Fig. 1.
clearer elastic consequence occurs when the flow stops, while in the case when slippage ceases all points of the medium will move in the opposite direction and the elastic strains fall to zero when the pressure drop is completely removed ( = 0). The analytical solution obtained may be useful in calculations of technological metal drawing operations (the numerical values of the constants given above are for steel), the transportation of heavy hydrocarbons etc., and serve as a basis for testing algorithms and numerical calculation programs. Note, finally, that we have used here the simplest relations for the friction, both dry and viscous. Corresponding calculations can be carried out for more complex friction laws. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
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Translated by R.C.G.