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Conical pipe envelope formation process
Journal of Materials Processing Technology 139 (2003) 164–169
Conical pipe envelope formation process F. Knap, R. Kruzel, M. Krakowiak∗ Department of Metallurgy and Materials Engineering, Institute of Modelling and Automation of Plastic Forming Processes, Technical University of Czestochowa, Armii Krajowej 19, 42-200 Czestochowa, Poland
Abstract Conical pipes belong to a group of economical profiles, which find broad application in various metal products. Until now conical pipes have been manufactured by the following methods: rolling on a two-high mill with impressions of varying depth on the rolls’ perimeter; forging in a lever swaging machine, forming in a combined process of drawing and rolling, and rolling on a skew rolling mill. The manufacturing of conical pipes belongs to complicated engineering processes, and the methods used until now do not allow the meeting of the demand for conical pipes—especially long ones with large cross-wise dimensions, e.g. pipes for lamp posts. This situation makes it reasonable to look for new processes with wider engineering possibilities when compared with the known methods known at present. It seems that the process of conical pipe envelope formation will enable the production of a wide assortment of conical pipes. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Conical pipe; Envelope formation process
1. Introduction In a conical pipe envelope formation process length-wise tensile stresses and circumferential and radial compressive stresses act on the deformed material, amongst which the circumferential stresses prevail. In the strain area, the stress condition causes a reduction of the outer diameter, an increase of wall thickness, and a slight pipe elongation. When plastically strained, the pipe undergoes unnecessary non-dilatational strains that are greater on the outer surface than on the inner one.
2. Laboratory testing of conical pipes forming with an envelope rolling method: the test stand The test stand (Fig. 1) to form conical pipes was installed on a lathe. The stand base, including the device body, was screwed down to the lathe slide. In the process a pipe is put in to rotation, and its radial straining take place through four rolls symmetrically arranged in relation to the deformed pipe (Fig. 2). The location of the rolls in relation to the pipe is determined by the conical surface of the fixing sleeve 1 connected with ∗ Corresponding author. E-mail address: [email protected] (M. Krakowiak).
the body 2 by a thread, so when the sleeve is set in to rotation, this causes its axial displacement, and simultaneously starts the radial shifting of rolls 4, and through that this motion makes it possible to receive a conical pipe. Conical pipe formation may occur with increasing or decreasing draft, and the pipe taper depends on the process kinematics conditions, i.e. the length-wise displacement speed of the forming rolls, and their radial shifting speed.
3. Process characteristics In the process of conical pipe envelope forming [1], an input cylindrical pipe, put in to rotation, is drawn between four rolls, symmetrically arranged in relation to the deformed pipe, and the roll generator is declined at some angle in relation to the deformed pipe, so gradual decreasing of the outer diameter takes place. During the process, a transition of a given point through a plastic strain valley is connected with multiple entering in a zone of direct roll impact, and the total strain is the sum of unit strains resulting from each roll impact on the deformed pipe. Pipe plastic strains that are expressed by reduction of the outer pipe diameter, and by length-wise and radial strains, take place as a result of direct impact of the forming rolls as well as of zones of non-contacted interaction on a boundary of the zones between forming rolls in which the elastic state prevails, and zones of plastic state.
0924-0136/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-0136(03)00215-2
F. Knap et al. / Journal of Materials Processing Technology 139 (2003) 164–169
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Fig. 1. Diagram of a test stand to roll conical pipes: 1, lathe slide; 2, rolling device; 3, worm gear; 4, gear drive motor; 5, base plate; 6, rolled pipe.
Fig. 2. Cross-section of the conical pipe envelope rolling device: 1, fixing sleeve; 2, body; 3, pipe; 4, roll; 5, mandrel. Designation of particular element motion during the process.
4. Conditions of tests conducting The tests were carried out on cooper pipes of diameter Ø13.5 mm × 0.75 mm. The plasticity limit of the pipe material was: σp = 210 MPa. The mean asymmetrical profile deviation from the mean line—Ra , that characterizes the outer surface condition, was Ra = 0.24 m, while the value for the inner surface was Ra = 0.28 m. The forming rolls (Fig. 3) were made of sintered carbides of G20 grade, and their active faces were polished. The fixing sleeve was made of quenched and tempered tool steel of NC10 grade. The pipe rotational speed was 280 rpm, and the device body displacement speed was 0.130–0.450 mm per revolution. The tests were carried out with the use of ŁT-4S bearing grease.
Fig. 3. Forming roll.
During the process, the length-wise force that acted on the deformed pipe was recorded through strain gauges. To determine the linear and non-dilatational strains in the plastic strain valley of the deformed pipe the process was stopped when the assumed pipe outer diameter change was achieved, and then the linear and non-dilatational strains were determined. Measurements of linear strains carried out in the circumferential, radial and length-wise directions (Fig. 4) showed that decreasing of pipe outer diameter is accompanied by an increase of the pipe wall thickness with slight pipe elongation.
5. Course and results of tests When beginning the process, firstly the pipe was put into rotation, and then the fixing sleeve was rotated together with a device providing length-wise displacement in relation to the deformed pipe. The process was conducted with a rotating fixing sleeve, both in the direction corresponding, as well as reversed, to the direction of the deformed pipe rotation. The tests were carried out with a roll length-wise speed within the bounds of 0.130–0.450 mm per revolution.
Fig. 4. Diagram of linear strain changes in the circumferential, radial and length-wise directions.
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Within the whole range of outer diameter, the decrease of the wall thickening took place monotonically as the pipe diameter decreased, so that in radial direction the strains are positive. Also the strains in the length-wise direction are positive, but their value is much less than for the radial strains. On the other hand, the strain in the circumferential direction is negative, as the pipe perimeter decreases. From among the three linear strains, the circumferential strain is the highest as far as absolute value is concerned; next the radial strain is slight less; with the length-wise strain being the smallest. The circumferential strains, which express the diameter change, and the radial strains, which express as wall thickness change, are the prevailing strains. At first approximation, it can be assumed that the length-wise strains are equal to zero, i.e. the pipe does not changes its length while it is being formed, and the outer diameter decrease is accompanied by wall thickness increase. Such a scheme of strains relates to the stress state that rules in plasticized zones. Length-wise stresses σ l , radial stresses σ r , and circumferential stresses σ θ l act on a deformed pipe. Length-wise stresses σ l cause elongation of the deformed material and its shortening in the radial and circumferential directions. Radial stresses have a similar effect, while circumferential stresses cause material twisting in the circumferential direction, and dimension increase in the radial direction, i.e. wall thickening. Conditions of stress equilibrium must be met between the radial and circumferential stresses that act in a pipe (Fig. 5). Projecting stresses that act on an elementary volume in the vertical direction: θ σr dϕrr cos ϕ l = σθ g0 l sin θ 0
When integrated and transformed: σr = σθ
g0 sin θ rr sin θ
(1)
When friction forces are taken into account the equilibrium condition assumes the form: θ (σr + σr µ sin ϕ)rr dϕ cos ϕ l = σθ g0 l sin θ 0
Hence σr =
σθ g0 sin θ rr sin θ + (1/4)µrr (1 − cos 2θ)
(2)
The above-mentioned relationships show the dependencies between radial and circumferential stresses on the pipe surface. Radial stress σ r on the inner surface is equal to zero, so it is proper to enter mean radial values against the wall thickness. Similarly as in a traditional process of pipe drawing [2], it was assumed that a pattern of radial stresses σ r is linear, and in this connection the mean radial stress σrm is equal to a half of the radial stress on outer pipe surface: σrm = 21 σr From the above equation, it may be seen that for a given value of the stress σ θ , the radial stresses σ rm are higher the higher is the wall-thickness-to-pipe-diameter ratio. Thus, when forming thick-walled pipes, the radial stresses are higher than in the course of forming thin-walled pipes. During conical-pipes forming the circumferential and length-wise stresses prevail. These stresses must meet a condition of plasticity. When using the condition of maximum tangential stresses: σl − σθ = βσp
(3)
Length-wise stresses σ l have a positive sign, while circumferential stresses σ θ have a negative sign. Thus if σ l is small, then σ θ is close to a value of plasticizing stresses and they determine the strain characteristics, and on the other hand when σ l is high then σ θ is small and stresses σ l will have a decisive influence on the strain pattern. Radial stresses will influence the situation to some extent, especially when the circumferential stresses σ θ are high. To analyze wall thickness changes, based on physical relations between strains and stresses [3–5]: ε˙ r − ε˙ m = 2G (σrm − σm )
where σm = mean stress
ε˙ θ − ε˙ m = 2G (σθ − σm )
(4) (5)
If taking into account: ε˙ r −
dεr , dt
ε˙ θ =
dεθ dt
(6)
and then dividing on both sides: Fig. 5. Diagram of stress pattern on valley section.
dεθ σrm − σm = dεθ σθ − σ m
(7)
F. Knap et al. / Journal of Materials Processing Technology 139 (2003) 164–169
At the same time the strain dεr , equal to dg/g, describes wall thickness changes in a given cross-section while dεθ , equal to dDr /Dr , describes a diameter change. Taking into account in Eq. (7), the relationship (1) between radial and circumferential stresses in which the expression sin θ/sin θ was designated as A, and the plasticity condition (3), and also entering a W coefficient that characterizes the length-wise stresses to plasticizing strain ratio W = σl /σp , upon being transformed: dg/g 2A(g/Dr )(β − W) + 2W − β = =n (8) dDr /Dr 2β − W − A(g/Dr )(β − W) The circumferential strain dDr /Dr always has a negative sign since the deformed pipe diameter decreases monotonically, while the radial strain, depending on the process realization conditions, may be positive, negative or equal to zero. Here the relative length-wise stress is the parameter that determines the character of wall thickness change. The W coefficient is contained in with in limits: 0 < W < 1. For relatively small values of W, the radial-strains-to-circumferential-strains ratio has a negative sign, which means that the wall thickness increases. For some value of the W coefficient a function that describes the strains ratio achieves a value equal to zero, which means that the wall thickness changes. Such a condition occur when the expression (8) numerator is equal to zero: g 2A (β − W) + 2W − β = 0 (9) Dr hence W=
2Aβ(g/Dr ) − β 2A(g/Dr ) − 2
(10)
This means that the value of the W coefficient for which the wall thickness does not change depends on the wallthickness-to-rolls-diameter ratio. For thin-walled pipes the coefficient value approaches a value equal to β/2, and when one assumes that β = 1.1 then the coefficient is equal to 0.55. For pipes with a relatively high-wall-thicknessto-diameter ratio, the value of the W coefficient for which the wall thickness does not change decreases, e.g. for 2g/Dr equal to 0.3 the W coefficient is equal to 0.423. Therefore, as the wall-thickness-to-roll-diameter ratio increases the W coefficient value, for which wall thickness does not change decreases. Thus the thickening range during plastic forming decreases. For high values of the W coefficient that depend on the wall-thickness-to-roll-diameter ratio, a function determining the radial-strain-to-circumferential-strain ratio has a positive sign, which means that wall thickness decreases. Thus if forming takes place with a relatively small W coefficient value then wall thickening will occur. On the other hand, if the W coefficient reaches high values then outer diameter decrease will occur initially with increasing wall thickness; then for some pipe diameter decrease it will achieve a maximum value; and then it will decrease. In conducted tests the n coefficient had a negative sign and changed
167
within limits of from −0.480 to −0.280, which means that pipe forming takes places with wall thickness increase, and at the same time its intensity decreases as the strains increase. Measurements carried out of the wall thickness revealed that in the tested range of outer diameter decrease, wall thickness increase took place. The authors failed to achieve high values of the n coefficient since when 30% decrease of outer diameter was reached, loss of pipe stability occurred. A value of limiting strain in the process of conical pipe envelope forming that amounts to approximately 30%, should be recognized as a high one compared to other pipe deforming processes, such as rolling on reducing mills, where the diameter decrease while passing one rolls set (two-high stands, or three-high stands) is of the order of 0.1. Higher strains are achieved during free pipe drawing, where the strain is of the order of 0.2. Thus the relatively high strain value achieved in the conical pipe envelope forming process is an essential positive characteristic of this new process of conical pipe forming. Wall thickness change depends also on wall-thicknessto-roll-diameter ratio. Higher values of the ratio diminish reduce the amount of thickening. Conducted tests demonstrated that during conical pipe envelope forming, apart from the desired linear strains, the deformed pipe undergoes unwanted non-dilatational strains in the three basic surfaces (Figs. 6 and 7). The presence of the strains testifies to the occurrence of tangential stresses in surfaces that are concentric in relation to the outer surface, in pipe longitudinal section and cross-section. The strain in concentric cylindrical surfaces is connected with the action of tangential stresses during forced pipe rotation with the occurrence of external torque. Its braking influence of forming rolls causes speed variation
Fig. 6. Diagram of three basic non-dilatational strains changes for the conical pipe outer surface.
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Fig. 7. Diagram of three basic non-dilatational strains changes for the conical pipe inner surface.
along the pipe length, which gives as a result non-dilatational strain in a cylindrical surface. The pipe receives higher non-dilatational strains on its outer surface. Non-dilatational strain in a pipe cross-section is connected with the decreasing influence of the forming rolls on the angular speed diversification over the pipe radius. In this case, again, the strains on the outer surface are higher than the strains on the inner surface. Similar regularities refer also to non-dilatational strains in the pipe longitudinal section. To characterize the non-dilatational share in the strain state, Blazynski [6] introduced a strain uselessness coefficient defined as the ratio of strain intensity including nondilatational strains, to strain intensity without non-dilatational strains: εHp ϕ= (11) εH Fig. 8 shows value of the coefficient for various diameter deformation values. It is seen in this figure that the coefficient is only a few percent higher than in the case of ideal strain. It is an essential characteristic of the conical pipe envelope forming process that says that the non-dilatational strains share in a strain tensor is relatively small. Thus plastic strain energy is used mainly for effective linear strain, while barely a few percent of the total energy goes for non-dilatational strain. In the experimental part of the work linear and nondilatational strains were determined and length-wise stresses acting in the input plane of the pipe from an area of plastic strain were determined, also. Based on the equilibrium condition of stresses that acts on an elementary ring, separated from strain area, and on the plasticity condition, circumferential and radial stresses were determined. The data enables the determination of the tangential stresses on the basis of
Fig. 8. Changes of ϕ coefficient for various diameters of conical pipe.
the physical connections between the stresses and strains in the plasticity state: σl − σm = 2G (˙εl − ε˙ m ),
σrm − σm = 2G (˙εr − ε˙ m ), γlr σθ − σm = 2G (˙εθ − ε˙ m ), τlr = 2G , 2 γrθ γθl τrθ = 2G , τθl = 2G (12) 2 2
When both sides of the equations that contain linear and circumferential strains speeds are divided by the plasticity modulus G and time differential dt is eliminated, stress
Fig. 9. Diagram of tangential stress values during conical pipe forming.
F. Knap et al. / Journal of Materials Processing Technology 139 (2003) 164–169
state components and one-at-a-time differential of linear and non-dilatational strains remain in the equations. The process of envelope forming is significantly characterized by the possibility to obtain relatively high total strains with relatively small elementary strains during one roll passage. The undergoing strains meet a monotony condition, and due to their very small value (0.00037), strain differentials may be substituted for by unitary strains. As a result of transformations: 2γθr (σθ − σm ), εθ 2γθl (σθ − σm ) τθl = εθ
τθr =
τrl =
2γrl (σθ − σm ), εθ
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compressive stresses act on deformed material, and at the same time the circumferential stresses prevail. The stress condition that prevails in the strain area causes reduction of outer diameter, increase of wall thickness, and slight pipe elongation. 2. Apart from the desired linear strains the pipe undergoes (unnecessary) non-dilatational strains. These strains are greater on the outer surface than on the inner surface.
References (13)
These equations allow determining the mean values of tangential stresses during pipe forming (Fig. 9). Conducted tests revealed that the non-dilatational strains are relatively small compared with the plasticizing stress values. Small values of the stresses are associated with small values of non-dilatational strains.
6. Conclusions 1. In the process of conical pipe envelope forming, tensile length-wise stresses and circumferential and radial
[1] F. Knap, R. Kruzel, Walcowanie obwiedniowe rur sto˙zkowych, Rudy Metale 8 (42) (1997) 338. [2] L. Sadok, M. Morawiecki, E. Wosiek, Przeróbka Plastyczna, Podstawy ´ ask, 1986. Teoretyczne, Katowice, Wydawnictwo Sl˛ [3] A.C. Atkins, R.M. Caddel, The incorporation of work hardening and redundant work in rod drawing analyses, Int. J. Mech. Sci. 10 (10) (1968) 15–28. [4] R.M. Caddel, A.C. Atkins, The influence of redundant work with drawing rods trough conical dies, Trans. ASME Paper No. 67—WA, Prod. 11. [5] R.W. Johnson, G.W. Rowe, Redundant work in drawing cylindrical stock, J. Inst. Met. 67 (1968) 97–105. [6] T.Z. Blazynski, Metal Forming—Tool Profiles and Flow, Macmillan, London, 1976.
Conical pipe envelope formation process F. Knap, R. Kruzel, M. Krakowiak∗ Department of Metallurgy and Materials Engineering, Institute of Modelling and Automation of Plastic Forming Processes, Technical University of Czestochowa, Armii Krajowej 19, 42-200 Czestochowa, Poland
Abstract Conical pipes belong to a group of economical profiles, which find broad application in various metal products. Until now conical pipes have been manufactured by the following methods: rolling on a two-high mill with impressions of varying depth on the rolls’ perimeter; forging in a lever swaging machine, forming in a combined process of drawing and rolling, and rolling on a skew rolling mill. The manufacturing of conical pipes belongs to complicated engineering processes, and the methods used until now do not allow the meeting of the demand for conical pipes—especially long ones with large cross-wise dimensions, e.g. pipes for lamp posts. This situation makes it reasonable to look for new processes with wider engineering possibilities when compared with the known methods known at present. It seems that the process of conical pipe envelope formation will enable the production of a wide assortment of conical pipes. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Conical pipe; Envelope formation process
1. Introduction In a conical pipe envelope formation process length-wise tensile stresses and circumferential and radial compressive stresses act on the deformed material, amongst which the circumferential stresses prevail. In the strain area, the stress condition causes a reduction of the outer diameter, an increase of wall thickness, and a slight pipe elongation. When plastically strained, the pipe undergoes unnecessary non-dilatational strains that are greater on the outer surface than on the inner one.
2. Laboratory testing of conical pipes forming with an envelope rolling method: the test stand The test stand (Fig. 1) to form conical pipes was installed on a lathe. The stand base, including the device body, was screwed down to the lathe slide. In the process a pipe is put in to rotation, and its radial straining take place through four rolls symmetrically arranged in relation to the deformed pipe (Fig. 2). The location of the rolls in relation to the pipe is determined by the conical surface of the fixing sleeve 1 connected with ∗ Corresponding author. E-mail address: [email protected] (M. Krakowiak).
the body 2 by a thread, so when the sleeve is set in to rotation, this causes its axial displacement, and simultaneously starts the radial shifting of rolls 4, and through that this motion makes it possible to receive a conical pipe. Conical pipe formation may occur with increasing or decreasing draft, and the pipe taper depends on the process kinematics conditions, i.e. the length-wise displacement speed of the forming rolls, and their radial shifting speed.
3. Process characteristics In the process of conical pipe envelope forming [1], an input cylindrical pipe, put in to rotation, is drawn between four rolls, symmetrically arranged in relation to the deformed pipe, and the roll generator is declined at some angle in relation to the deformed pipe, so gradual decreasing of the outer diameter takes place. During the process, a transition of a given point through a plastic strain valley is connected with multiple entering in a zone of direct roll impact, and the total strain is the sum of unit strains resulting from each roll impact on the deformed pipe. Pipe plastic strains that are expressed by reduction of the outer pipe diameter, and by length-wise and radial strains, take place as a result of direct impact of the forming rolls as well as of zones of non-contacted interaction on a boundary of the zones between forming rolls in which the elastic state prevails, and zones of plastic state.
0924-0136/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-0136(03)00215-2
F. Knap et al. / Journal of Materials Processing Technology 139 (2003) 164–169
165
Fig. 1. Diagram of a test stand to roll conical pipes: 1, lathe slide; 2, rolling device; 3, worm gear; 4, gear drive motor; 5, base plate; 6, rolled pipe.
Fig. 2. Cross-section of the conical pipe envelope rolling device: 1, fixing sleeve; 2, body; 3, pipe; 4, roll; 5, mandrel. Designation of particular element motion during the process.
4. Conditions of tests conducting The tests were carried out on cooper pipes of diameter Ø13.5 mm × 0.75 mm. The plasticity limit of the pipe material was: σp = 210 MPa. The mean asymmetrical profile deviation from the mean line—Ra , that characterizes the outer surface condition, was Ra = 0.24 m, while the value for the inner surface was Ra = 0.28 m. The forming rolls (Fig. 3) were made of sintered carbides of G20 grade, and their active faces were polished. The fixing sleeve was made of quenched and tempered tool steel of NC10 grade. The pipe rotational speed was 280 rpm, and the device body displacement speed was 0.130–0.450 mm per revolution. The tests were carried out with the use of ŁT-4S bearing grease.
Fig. 3. Forming roll.
During the process, the length-wise force that acted on the deformed pipe was recorded through strain gauges. To determine the linear and non-dilatational strains in the plastic strain valley of the deformed pipe the process was stopped when the assumed pipe outer diameter change was achieved, and then the linear and non-dilatational strains were determined. Measurements of linear strains carried out in the circumferential, radial and length-wise directions (Fig. 4) showed that decreasing of pipe outer diameter is accompanied by an increase of the pipe wall thickness with slight pipe elongation.
5. Course and results of tests When beginning the process, firstly the pipe was put into rotation, and then the fixing sleeve was rotated together with a device providing length-wise displacement in relation to the deformed pipe. The process was conducted with a rotating fixing sleeve, both in the direction corresponding, as well as reversed, to the direction of the deformed pipe rotation. The tests were carried out with a roll length-wise speed within the bounds of 0.130–0.450 mm per revolution.
Fig. 4. Diagram of linear strain changes in the circumferential, radial and length-wise directions.
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Within the whole range of outer diameter, the decrease of the wall thickening took place monotonically as the pipe diameter decreased, so that in radial direction the strains are positive. Also the strains in the length-wise direction are positive, but their value is much less than for the radial strains. On the other hand, the strain in the circumferential direction is negative, as the pipe perimeter decreases. From among the three linear strains, the circumferential strain is the highest as far as absolute value is concerned; next the radial strain is slight less; with the length-wise strain being the smallest. The circumferential strains, which express the diameter change, and the radial strains, which express as wall thickness change, are the prevailing strains. At first approximation, it can be assumed that the length-wise strains are equal to zero, i.e. the pipe does not changes its length while it is being formed, and the outer diameter decrease is accompanied by wall thickness increase. Such a scheme of strains relates to the stress state that rules in plasticized zones. Length-wise stresses σ l , radial stresses σ r , and circumferential stresses σ θ l act on a deformed pipe. Length-wise stresses σ l cause elongation of the deformed material and its shortening in the radial and circumferential directions. Radial stresses have a similar effect, while circumferential stresses cause material twisting in the circumferential direction, and dimension increase in the radial direction, i.e. wall thickening. Conditions of stress equilibrium must be met between the radial and circumferential stresses that act in a pipe (Fig. 5). Projecting stresses that act on an elementary volume in the vertical direction: θ σr dϕrr cos ϕ l = σθ g0 l sin θ 0
When integrated and transformed: σr = σθ
g0 sin θ rr sin θ
(1)
When friction forces are taken into account the equilibrium condition assumes the form: θ (σr + σr µ sin ϕ)rr dϕ cos ϕ l = σθ g0 l sin θ 0
Hence σr =
σθ g0 sin θ rr sin θ + (1/4)µrr (1 − cos 2θ)
(2)
The above-mentioned relationships show the dependencies between radial and circumferential stresses on the pipe surface. Radial stress σ r on the inner surface is equal to zero, so it is proper to enter mean radial values against the wall thickness. Similarly as in a traditional process of pipe drawing [2], it was assumed that a pattern of radial stresses σ r is linear, and in this connection the mean radial stress σrm is equal to a half of the radial stress on outer pipe surface: σrm = 21 σr From the above equation, it may be seen that for a given value of the stress σ θ , the radial stresses σ rm are higher the higher is the wall-thickness-to-pipe-diameter ratio. Thus, when forming thick-walled pipes, the radial stresses are higher than in the course of forming thin-walled pipes. During conical-pipes forming the circumferential and length-wise stresses prevail. These stresses must meet a condition of plasticity. When using the condition of maximum tangential stresses: σl − σθ = βσp
(3)
Length-wise stresses σ l have a positive sign, while circumferential stresses σ θ have a negative sign. Thus if σ l is small, then σ θ is close to a value of plasticizing stresses and they determine the strain characteristics, and on the other hand when σ l is high then σ θ is small and stresses σ l will have a decisive influence on the strain pattern. Radial stresses will influence the situation to some extent, especially when the circumferential stresses σ θ are high. To analyze wall thickness changes, based on physical relations between strains and stresses [3–5]: ε˙ r − ε˙ m = 2G (σrm − σm )
where σm = mean stress
ε˙ θ − ε˙ m = 2G (σθ − σm )
(4) (5)
If taking into account: ε˙ r −
dεr , dt
ε˙ θ =
dεθ dt
(6)
and then dividing on both sides: Fig. 5. Diagram of stress pattern on valley section.
dεθ σrm − σm = dεθ σθ − σ m
(7)
F. Knap et al. / Journal of Materials Processing Technology 139 (2003) 164–169
At the same time the strain dεr , equal to dg/g, describes wall thickness changes in a given cross-section while dεθ , equal to dDr /Dr , describes a diameter change. Taking into account in Eq. (7), the relationship (1) between radial and circumferential stresses in which the expression sin θ/sin θ was designated as A, and the plasticity condition (3), and also entering a W coefficient that characterizes the length-wise stresses to plasticizing strain ratio W = σl /σp , upon being transformed: dg/g 2A(g/Dr )(β − W) + 2W − β = =n (8) dDr /Dr 2β − W − A(g/Dr )(β − W) The circumferential strain dDr /Dr always has a negative sign since the deformed pipe diameter decreases monotonically, while the radial strain, depending on the process realization conditions, may be positive, negative or equal to zero. Here the relative length-wise stress is the parameter that determines the character of wall thickness change. The W coefficient is contained in with in limits: 0 < W < 1. For relatively small values of W, the radial-strains-to-circumferential-strains ratio has a negative sign, which means that the wall thickness increases. For some value of the W coefficient a function that describes the strains ratio achieves a value equal to zero, which means that the wall thickness changes. Such a condition occur when the expression (8) numerator is equal to zero: g 2A (β − W) + 2W − β = 0 (9) Dr hence W=
2Aβ(g/Dr ) − β 2A(g/Dr ) − 2
(10)
This means that the value of the W coefficient for which the wall thickness does not change depends on the wallthickness-to-rolls-diameter ratio. For thin-walled pipes the coefficient value approaches a value equal to β/2, and when one assumes that β = 1.1 then the coefficient is equal to 0.55. For pipes with a relatively high-wall-thicknessto-diameter ratio, the value of the W coefficient for which the wall thickness does not change decreases, e.g. for 2g/Dr equal to 0.3 the W coefficient is equal to 0.423. Therefore, as the wall-thickness-to-roll-diameter ratio increases the W coefficient value, for which wall thickness does not change decreases. Thus the thickening range during plastic forming decreases. For high values of the W coefficient that depend on the wall-thickness-to-roll-diameter ratio, a function determining the radial-strain-to-circumferential-strain ratio has a positive sign, which means that wall thickness decreases. Thus if forming takes place with a relatively small W coefficient value then wall thickening will occur. On the other hand, if the W coefficient reaches high values then outer diameter decrease will occur initially with increasing wall thickness; then for some pipe diameter decrease it will achieve a maximum value; and then it will decrease. In conducted tests the n coefficient had a negative sign and changed
167
within limits of from −0.480 to −0.280, which means that pipe forming takes places with wall thickness increase, and at the same time its intensity decreases as the strains increase. Measurements carried out of the wall thickness revealed that in the tested range of outer diameter decrease, wall thickness increase took place. The authors failed to achieve high values of the n coefficient since when 30% decrease of outer diameter was reached, loss of pipe stability occurred. A value of limiting strain in the process of conical pipe envelope forming that amounts to approximately 30%, should be recognized as a high one compared to other pipe deforming processes, such as rolling on reducing mills, where the diameter decrease while passing one rolls set (two-high stands, or three-high stands) is of the order of 0.1. Higher strains are achieved during free pipe drawing, where the strain is of the order of 0.2. Thus the relatively high strain value achieved in the conical pipe envelope forming process is an essential positive characteristic of this new process of conical pipe forming. Wall thickness change depends also on wall-thicknessto-roll-diameter ratio. Higher values of the ratio diminish reduce the amount of thickening. Conducted tests demonstrated that during conical pipe envelope forming, apart from the desired linear strains, the deformed pipe undergoes unwanted non-dilatational strains in the three basic surfaces (Figs. 6 and 7). The presence of the strains testifies to the occurrence of tangential stresses in surfaces that are concentric in relation to the outer surface, in pipe longitudinal section and cross-section. The strain in concentric cylindrical surfaces is connected with the action of tangential stresses during forced pipe rotation with the occurrence of external torque. Its braking influence of forming rolls causes speed variation
Fig. 6. Diagram of three basic non-dilatational strains changes for the conical pipe outer surface.
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Fig. 7. Diagram of three basic non-dilatational strains changes for the conical pipe inner surface.
along the pipe length, which gives as a result non-dilatational strain in a cylindrical surface. The pipe receives higher non-dilatational strains on its outer surface. Non-dilatational strain in a pipe cross-section is connected with the decreasing influence of the forming rolls on the angular speed diversification over the pipe radius. In this case, again, the strains on the outer surface are higher than the strains on the inner surface. Similar regularities refer also to non-dilatational strains in the pipe longitudinal section. To characterize the non-dilatational share in the strain state, Blazynski [6] introduced a strain uselessness coefficient defined as the ratio of strain intensity including nondilatational strains, to strain intensity without non-dilatational strains: εHp ϕ= (11) εH Fig. 8 shows value of the coefficient for various diameter deformation values. It is seen in this figure that the coefficient is only a few percent higher than in the case of ideal strain. It is an essential characteristic of the conical pipe envelope forming process that says that the non-dilatational strains share in a strain tensor is relatively small. Thus plastic strain energy is used mainly for effective linear strain, while barely a few percent of the total energy goes for non-dilatational strain. In the experimental part of the work linear and nondilatational strains were determined and length-wise stresses acting in the input plane of the pipe from an area of plastic strain were determined, also. Based on the equilibrium condition of stresses that acts on an elementary ring, separated from strain area, and on the plasticity condition, circumferential and radial stresses were determined. The data enables the determination of the tangential stresses on the basis of
Fig. 8. Changes of ϕ coefficient for various diameters of conical pipe.
the physical connections between the stresses and strains in the plasticity state: σl − σm = 2G (˙εl − ε˙ m ),
σrm − σm = 2G (˙εr − ε˙ m ), γlr σθ − σm = 2G (˙εθ − ε˙ m ), τlr = 2G , 2 γrθ γθl τrθ = 2G , τθl = 2G (12) 2 2
When both sides of the equations that contain linear and circumferential strains speeds are divided by the plasticity modulus G and time differential dt is eliminated, stress
Fig. 9. Diagram of tangential stress values during conical pipe forming.
F. Knap et al. / Journal of Materials Processing Technology 139 (2003) 164–169
state components and one-at-a-time differential of linear and non-dilatational strains remain in the equations. The process of envelope forming is significantly characterized by the possibility to obtain relatively high total strains with relatively small elementary strains during one roll passage. The undergoing strains meet a monotony condition, and due to their very small value (0.00037), strain differentials may be substituted for by unitary strains. As a result of transformations: 2γθr (σθ − σm ), εθ 2γθl (σθ − σm ) τθl = εθ
τθr =
τrl =
2γrl (σθ − σm ), εθ
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compressive stresses act on deformed material, and at the same time the circumferential stresses prevail. The stress condition that prevails in the strain area causes reduction of outer diameter, increase of wall thickness, and slight pipe elongation. 2. Apart from the desired linear strains the pipe undergoes (unnecessary) non-dilatational strains. These strains are greater on the outer surface than on the inner surface.
References (13)
These equations allow determining the mean values of tangential stresses during pipe forming (Fig. 9). Conducted tests revealed that the non-dilatational strains are relatively small compared with the plasticizing stress values. Small values of the stresses are associated with small values of non-dilatational strains.
6. Conclusions 1. In the process of conical pipe envelope forming, tensile length-wise stresses and circumferential and radial
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