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Analytical process design for interference-fit joining of rectangular profiles
Journal of Materials Processing Tech. 276 (2020) 116391
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Analytical process design for interference-fit joining of rectangular profiles F. Weber , M. Müller, P. Haupt, S. Gies, M. Hahn, A.E. Tekkaya ⁎
T
Institute of Forming Technology and Lightweight Components, TU Dortmund University, Baroper Straße 303, 44227, Dortmund, Germany
ARTICLE INFO
ABSTRACT
Associate Editor: Z Cui
The classical joining of round tubes by die-less hydroforming is extended to joining of rectangular profiles. A new analytical approach for the calculation of required process parameters is developed. It allows the calculation of the interference pressure p and the required fluid pressure range given by a lower and upper limit ( pi,min and pi,max ). These limits are the result of the deformation state of the inner and outer joining partner. The minimum pressure, where plastic deformation of the inner joining partner occurs first, characterizes the value at which the joint formation starts. The maximum pressure is the value, at which plastic deformation of the outer joining partner occurs. The analytical model is validated by experimental studies where two rectangular aluminum profiles (EN AW-6060) with different heat treatment conditions were joined. Above pi,max , the increase of the achievable joint strength with respect to the internal pressure is not significant anymore. The maximum deviation between the analytically and experimentally determined maximum fluid pressure pi,max is about 12% in case of a wall thickness of 3 mm and 11% in case of a wall thickness of 5 mm.
Keywords: Joining by forming Die-less hydroforming Intereference fit Profiles with rectangular cross-sectional shapes Analytical prediction
1. Introduction In the automotive industry, lightweight strategies like spaceframe structures are used (Barnes and Pashby, 2000) for well-known reasons. Lightweighting is often realized by the use of lightweight materials. The combination of two different materials leads to multi-material design. This strategy requires new joining technologies to create multi-material parts. In recent years it has been shown that joining by die-less hydroforming (DHF) is a suitable method for joining components with circular cross-sections such as camshafts (Garzke, 2001) and multi-layer pipes (Liu et al., 2004). The main advantage besides the joining of dissimilar materials is the elimination of additional joining elements, e.g. bolts, rivets or screws (Mori et al., 2013). Simple process management, the possibility of easy monitoring and the lack of heat input motivate the application of this joining technology. As depicted in Fig. 1a, the joint can be achieved by either force-fit, form-fit as well as a superposition of both mechanisms. According to Garzke (2001), the process can be divided into three main steps as displayed in Fig. 1b. In the first step, the joining partners, consisting of the outer and inner tube, are aligned in a shaft to hub configuration and the joining tool is placed inside the inner tube, underneath the joining zone. After that, a pressurized fluid (water or hydraulic oil) is applied into the gap between the tool and the inner tube. The seals limit the axial length of the pressurization area. As the pressure increases, the joining partners
⁎
begin to expand. Plastic deformation of the inner tube occurs when the inner stress reaches the tube’s initial yield strength Y0 (Yokell, 1992). A relief of the fluid pressure leads to the elastic recovery of both joining partners. The irreversible deformation of the inner tube prevents the full springback of the elastically expanded outer ring, which leads to a remaining interference pressure in the contact area. The interference pressure will increase with higher fluid pressure until an upper limit is reached. At this limit, a further increase in the fluid pressure leads to no significant increase in joint strength since the outer ring is plastically deformed and the maximum springback is reached (Marré, 2009). The strength of the joint can futher be increased by form-fit joining. In formfit joining, the material bulges into the provided grooves on the inside of the ring, which leads to an undercut (Podhorsky and Krips, 1990). First mentioned by Jantscha (1929); Podhorsky and Krips (1990) brought the process into industrial application for the production of tube heat exchangers. For this so-called tube-to-tube sheet joints, designing rules were given by Osweiller (1988). For his investigations, he created pipe to pipe-plate joints by an internal high pressure. There are investigations and physical models, describing the bulging process of the inner tube into the groove (Gies et al., 2012) to calculate the working fluid pressure and the influence of the elastic springback in the case of form-fit joining (Gies et al., 2013). Garzke (2001) investigated the behavior of joints under torsional loads and developed a physical model for the calculation of an average interference pressure. The model was combined with an empirical approach to increase its
Corresponding author. E-mail address: [email protected] (F. Weber).
https://doi.org/10.1016/j.jmatprotec.2019.116391 Received 10 January 2019; Received in revised form 10 August 2019; Accepted 9 September 2019 Available online 10 September 2019 0924-0136/ © 2019 Elsevier B.V. All rights reserved.
Journal of Materials Processing Tech. 276 (2020) 116391
F. Weber, et al.
Nomenclature
ML Elastic limit moment MY Yield moment p Inferference pressure pi , Fluid pressure pi,min , pi,max Minimum / maximum fluid pressure pi,target Experimental target fluid pressure Q Lateral force of the inner / outer joining partner q Line load of the inner / outer joining partner t Wall-thickness of the inner / outer joining partner Elastic stress across the beam’s cross-section xx,el Elastic-plastic stress across the beam´s cross-section xx,elpl Yield stress Y Initial yield stress Y0 µ Coulumb friction coefficient v lr Adhesion coefficient w Deflection of the beam wconstraint Elastic deformation constraint wdeformed Remaining deflection after holding time w loaded wunloaded Deflection in loaded / unloaded condition wmax Deflection at target fluid pressure wspringback Deflection due to springback wsum Remaining deflection y Coordinate in direction of profile length x Coordinate in direction of profile width x1, x2 Coordinate of transition from elastic to elastic-plastic domain xq Initial coordinate from where the partial line load q is acting z Coordinate in direction of profile height Profile height where elastic-plastic deformation occurs zY
Symbol/meaning1
A a b c cL c unloaded cY D Eel Epl Eelpl ¯ xx
el pl
F Fpl0.01 FAX Feq f h Iy L Li LqI+qO lf M Melpl
Contact surface Axial length of the inner / outer joining partner Width of the beam´s cross-section Curvature Elastic limit curvature Unloading curvature Yield curvature Diameter of the O-ring Young´s modulus of the inner / outer joining partner Plastic modulus of the inner / outer joining partner Elastic-plastic modulus Equivalent reference strain Strain in the beam´s cross-section Elastic elongation Plastic elongation Spring force of the inner / outer joining partner Axial connection strength Pull-out force Total spring force Deflection of local unloading Height of the beam´s cross-section Moment of inertia Length of the beam’s cross-section Length of the inner side of the inner / outer joining partner Length of the partial line load qI + qO , Length of the joining zone Bending moment of the inner / outer joining partner Moment in elastic-plastic area
accuracy in case of thick-walled outer joining partners. Grünendick (2004) published a model for the calculation of the interference pressure, taking into account the deformation in the radial and axial direction and investigated the connection behavior for oscillating loads. Marré (2009) developed an analytical approach for interference fit joining, in which he derived equations to calculate the required process variables such as minimum and maximum fluid pressure. Halle’s (2012) approach also allows calculations for interference fit joints. The difference to the other approaches is that the outer joining partner is thickwalled with a rotationally symmetric joining zone and a non-rotationally symmetric mass allocation. An adapted mass allocation can be found, for example, in lightweight optimized drive components. An extension of the process limits in terms of the geometry of the joining partners is given by Müller et al. (2018) for joining by hydraulic expansion of joining partners with oval cross-sectional shapes. Both et al. (2011) proved in experimental investigations that the process is also suitable for rectangular profiles. In their investigations, they used a tool as shown in Fig. 2, which was developed and patented by Marré et al. (2013). The process principle is identical to the joining of rotationally symmetric components whereas the tool design is different. The prototype tool has two seal concepts for the joining zones, a discontinous and a continuous one around the profile. Discontinuous joining zones (see Fig. 2a) are created by the O-ring-seals on each outer surface of the hydro probe where the pressurized fluid is applied. Continuous joining zones (see Fig. 2b) are made via two circumferential grooves for elastomer seals. Müller et al. (2016) published results of experimental and numerical investigations of both concepts. They state that, by assuming elastic behavior
of the outer joining partner, there is no significant difference between the two joining zone concepts regarding the internal stress distribution. The numerical results showed the same springback characteristic for both concepts. Therefore, the remaining interference pressure, which correlates with the achievable joint strength, is nearly the same. For joining rectangular profiles (see Fig. 3a and c), existing physical models cannot be used because they implicate a different springback behavior which is only valid for axisymmetric tubes where the springback is a result of the circumferential stress. Since springback in rectangular profiles occurs because of the bending stresses, a new approach has to be developed to determine pi,min , pi,max and p . Fig. 3b displays the loading characteristic with the main process parameters, including the deflection w of the profiles in the loaded and unloaded condition. Based on this loading situation, an analytical model is derived and presented in the next section. Section three describes the experimental methodology for the corresponding validation. The analytical and experimental results are then compared in section four. 2. Analytical model 2.1. Model assumptions A model is developed for the joining of profiles with squared crosssections employing the discontinuous sealing concept (Fig. 2a). Therefore, the profile’s cross-section is divided into four parts. Each side is considered as a clamped beam under a partial line load qI + qO , which represents the sum of the specific loads of the inner and outer beam, as shown in Fig. 4a. The length of the inner and outer beam is LI,i and LO,i respectively. The length of the line load in Eq. (1) is defined as the O-rings diameter.
1 In this arcticle the index I is used for the inner joining partner whereas O is used for variables concerning the outer joining partner.
LqI+qO = D 2
(1)
Journal of Materials Processing Tech. 276 (2020) 116391
F. Weber, et al.
Fig. 1. a) Joining mechanisms and b) process sequence for joining by die-less hydroforming (DHF) (Garzke, 2001).
Fig. 2. Prototype tool concepts (Marré, 2013) a) Discontinuous joining zone b) Continuous joining zone.
Due to the symmetry of the load and the boundary conditions of the clamped sides (w = w = 0 ), the analysis of one half of a beam is sufficient. The beams, representing the inner and outer joining partner, are in contact. This will be incorporated by considering them as elasticlinear plastic springs in parallel connection coupled by the equal deflection w of their midpoints as given in Eq. (2) (compare Fig. 5a).
wI
L L = wO 2 2
2.2. Extension of the Euler-Bernoulli beam theory 2.2.1. Elastic and elastic-plastic bending behavior For purely linear-elastic beams, the bending moment M (x ) , as a function of the curvature c (x ) = w (x ) , the moment of inertia Iy and the Young´s modulus Eel are related by Eq. (3).
M (x ) = (2)
Eel Iy c (x )
(3)
The product Eel Iy is called elastic bending stiffness. In the Euler-Bernoulli bending theory of elastic beams, it is assumed that the strain = xx and accordingly the stress = xx = Eel xx is linearly distributed over the cross-section (Fig. 6). The deformation of the beam under a line force q (x ) is represented by the deflection w (x ) , which can be calculated through integration of the governing differential Eq. (4) by taking into account known boundary conditions.
A linear elastic spring force is calculated as the product of the spring constant and the deflection. The total force Feq in parallel connection is the sum of the spring forces. For the two beams it is the sum of the line loads, qI and qO. In case of pure elasticity, both springs unload to zero if the load Feq is reduced to zero. However, for an elastic-linear plastic behavior of the springs, it is possible that an internal force arises during unloading. This is qualitatively sketched in Fig. 5b. If the load qI + qO on the two beams in contact is reduced to zero, an internal pressure p between them can be observed. For a quantitative analysis of this situation, the Euler-Bernoulli beam theory has to be extended to elastoplasticity.
Eel Iy w '''' (x ) = q (x )
(4)
The strain within a material fiber is proportional to the distance z from the neutral fiber and the curvature c (x ) , as seen in Eq. (5). 3
Journal of Materials Processing Tech. 276 (2020) 116391
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Fig. 3. a) Application case: Squared profiles b) Aims of the analytical approach for interference fit c) Lightweight frame structure of the Collborative Research Center SFB TR10.
=
xx (x ,
z) =
c z=
(5)
z w''(x )
The displayed elastic-plastic equations derive similar to the elastic equation but with the elastic plastic modulus Eelpl and by moving the equation by the value of ±zY along the z -axis and ± Y0 along the xx -axis. Integration of the bilinear function yields the bending moment as a function of curvature. Observing the symmetry of the stress distribution, the elastic-plastic bending moment is given by Eq. (9).
The stress distribution for linear elasticity is then calculated by Eq. (6). xx (x ,
z) =
Eel z w '' (x ) =
Eel z c
(6)
In order to extend these ideas to elastoplasticity, the assumption of a linear strain distribution is retained; but the stress distribution is replaced by the bilinear approximation for linear isotropic hardening with an elastic-plastic modulus in Eq. (7), where Eel = el is the stress-strain ratio in the elastic region and Epl = is the ratio in the pl plastic region.
Eelpl
=
Eel
+
Epl
E
= elpl
Melpl = 2 b
Y0
Eel c
z dz =
zY 0
xx (z )
z dz +
h /2 zY
xx (z )
z dz. (9)
(7)
Melpl (c ) =
The elastic and elastic-plastic stress distribution is illustrated in Fig. 7 for a negative value of the curvature. With the initial flow stress Y0 , the coordinate limiting the elastic region is given by Eq. (8).
zY =
xx (z )
For negative values of c the result of the calculation is given by Eq. (10).
Eel Epl Eel + Epl
h /2 0
Eel Eel + Epl
3
Iy h
Y
b
3 1 Y 3 Eel 2 c 2
Epl Iy c
(10)
Substituion of Iy for the given cross-section in Fig. 6 yields Eq. (11).
Melpl (c ) =
(8)
Eel b Eel + Epl
Y
4
h2
3 1 Y 3 Eel 2 c 2
Epl
h3 c 12
For positive values of c the same calculation yields Eq. (12).
Fig. 4. a) Model assumptions regarding the geometrical aspects b) Free body diagram. 4
(11)
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Fig. 5. a) Analogous spring model b) Schematic loading-unloading curve of elastic-plastic springs.
Melpl (c ) =
Eel Eel + Epl
3
Iy
Y
h
3 1 Y 3 Eel2 c 2
+b
Melpl ( ± c ) =
Epl Iy c
(12)
h h , i. e. 2 2
MY =
Y0
(13)
Eel |c|
2 Y0 Eel h
cY = (14)
b h2 6
Y0
.
Melpl (c ) =
Eel Eel + Epl
Iy
Melpl (c ) =
Eel Eel + Epl
=
3
Iy h
cL b
Y
h 3
c
Y
3
1 3 Eel 2 c 2 Y
+b
Y
3
3 Eel
2
1 c2
Epl Iy c
for c
cY ]
[c Y , [
(17)
(18)
MY 3 Y = (Eel Iy ) (Eel + Epl ) h
(19)
cY
c
cY cY cY
(20)
2.2.2. Loading cases and material states The material state of a beam depends on the bending moment caused by the external load. In Fig. 10, different possible states of loading of the left half of the beam are shown qualitatively. For each state, an external load q is applied from x q to L 2 . Initially, the beam deforms elastically for an increasing load, as seen in Fig. 10a, until the bending moment M (x = 0) approaches the value MY . As the external load increases, an elastic-plastic deformation arises at the clamping point. Thus, an expanding region of elastic-plastic behavior occurs within the area between x = 0 and x = x1, (see Fig. 10b). This is valid until a second elastic-plastic area arises starting from the midpoint x = L 2 . A further increase of the loading q leads to a second
Epl Iy c for c
c L. (16) The function M (c ) is depicted in Fig. 8 (solid line). There is a smooth transition from elastic to elastic-plastic states. For the further procedure, it is convenient to replace this function by its asymptotic approximation (Fig. 8, dashed line). This can be done by 3
,
(compare Fig. 9) The applied approach leads to a bilinear approximation of the moment-curvature relation. This facilitates an integration of the beam´s deflection curve.
M (c ) cL
Y
M (c ) MY Eelpl l y (c + cY ) if c MY Eelpl l y (c c Y ) if c
Considering this, a sectionally defined function M (c ) is given by Eq. (16).
Eel Iy c for
3 Iy Eel (Eel + Epl ) h
Eel l y c if
(15)
Mel (c ) =
]
The value of ML indicates plastsicization at the outer fiber of the beam, whereas the beam is fully plastic at MY . The elastic-plastic bending properties are then represented by Eq. (20).
Substituing Eq. (14) into Eqs. (10) and (12) respectively leads to the elastic limit moment ML in Eq. (15).
ML (c L ) =
c Y )for c
and
This gives the elastic limit curvature cL according to Eq. (14).
cL =
Eelpl Iy (c
with
The elastic description of the beam is valid for
|zY|
MY
simply neglecting the nonlinear term Y 2 2 for Melpl (c ) in Eq. (16). 3 Eel c This leads to the new definition in Eq. (17), 1
Fig. 6. a) Cross-section of the beam b) Linear stress distribution over the beams cross-section. 5
Journal of Materials Processing Tech. 276 (2020) 116391
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Fig. 7. Elastic and elastic plastic stress distribution over the beams cross-section (for c < 0).
Each general solution depends on four integration constants. The 16 } ) can be deconstants (Cj, Ej, Dj and Fj for {j | 1 j 4 j termined from the boundary conditions at the beam ends and the continuity requirements for the deflection w , the slope w = , the bending moment M and the lateral force Q at the transition points. 2.2.3. Resulting load-deflection relationship The objective of the following analysis is to establish a characteristic relation between the intensity of the external load q and the deflection
( )
of the beam’s midpoint w L 2 . Therefore, the boundary value problem given in Fig. 11 has to be solved in four steps:
1.) Elastic solution for the whole beam (Fig. 10a). Increase of the load q from 0 to the value of the yield moment MY at x = 0. This defines the first part of the characteristic curve q (w1) . This part is linear in contrast to the following parts. 2.) Solution of the situation in Fig. 10b. The determination of the transition point x = x1 between elastic-plastic and elastic behavior is part of the problem (root of a polynomial). As the load q is raised incrementally, x1 increases step by step, whereas the bending moment M L 2 increases simultaneously up to MY . This determines a point q (w2) of the characteristic curve. 3.) Solution of the situation of Fig. 10c. Now there are two transition points x1 and x2 to be calculated from M1 (x1, x2) = MY and M2 (x1, x2) = MY (roots of two polynomials depending on two variables). This calculation is repeated with increasing values of q, and stopped before x1 becomes larger than x q . This defines the last point q (w3) of the characteristic curve. 4.) Solution for unloading ((w4)= 0). This is addressed in more detail hereafter.
Fig. 8. Function of the bending moment M (c) .
( )
The characteristic curve for the loading and unloading of one spring (parameter values in Table 1) is sketched in Fig. 12. The listed material parameters ( Y0 , Eel , Epl ) of the inner beam in Table 1, as well as the below mentioned parameters of the outer beam, are determined by uniaxial tensile tests as laid down in DIN EN ISO 6892. Note that the shape of the curve between w1 and w3 is only indicated by straight lines, whereas it is nonlinear in reality. However, the values q (w1) , q (w2) and q (w3) are exact. The distribution of the bending moment is different from zero in the unloaded configuration q (w4 )= 0, because the support of the beam is statically undetermined. In order to get this distribution (and springback deflection w4 with q (w4)= 0), the principle of local unloading is employed, i.e. the bending moment is reduced at every point to zero. This produces a change in curvature, c unloaded (x ) , as visualized in general in Fig. 13. This change in curvature is the second derivative f (x ) of a function f (x ) , which can be interpreted to be the springback deflection of the unloaded beam. This curvature reduces to zero within the elastic
Fig. 9. Elastic-plastic bending properties as a function of curvature.
area with elastic-plastic material behavior from x2 to L 2 (see Fig. 10c). For each situation, a separate differential equation has to be solved. In case of purely elastic bending (Fig. 10a), the beam has to be divided into two areas. The area near to the clamped side is unloaded, followed by a loaded zone. The case of Fig. 10b is divided into three domains. At the clamped side, there is an unloaded elastic-plastic region (small bending stiffness), followed by a second unloaded but elastic region (large bending stiffness). The remaining region is also just elastic but loaded. The procedure for solving the differential equation is exemplarily shown in detail for the situation shown in Fig. 10c. One symmetric half of the beam is divided into four domains. As seen in Fig. 11, each domain is characterized by its load case (loaded/unloaded) and its material state, represented by a bending stiffness (elastic/elastic-plastic). 6
Journal of Materials Processing Tech. 276 (2020) 116391
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Fig. 10. Load cases: (a) Pure elastic bending (b) Elastic-plastic bending (c) Elastic-plastic bending in two domains.
Fig. 11. Differential equations for a beam with elastic-plastic domains.
regions, whereas in the plastic regions it follows from the elastic recovery. Thus, the springback is defined as follows in Eq. (21) to Eq. (24) for the regions 1 to 4:
c unloaded,1 =
M1 (x ) = f1 ''(x ) (Eel Iy )
c unloaded,3 = 0 = f3 ''(x ) c unloaded,4 =
(21)
(24)
This defines the curve f (x ) by integration. In doing so, the initial conditions f (0) = f (0)= 0 have to be considered as well as the continuity requirement of f and f at the transition points. For the example
(22)
c unloaded,2 = 0 = f2 ''(x )
M4 (x ) = f4 ''(x ) (Eel Iy )
(23)
Table 1 Parameter values. Bending situation
q [N/mm]
w I [mm]
Material
Elastic (Fig. 11a): q(w1) Elastic-plastic (Fig. 11b): q(w2) Elastic-plastic (Fig. 11c): q(w3) Unloading q (w4 )
26.24 33.05 51.13 0
0.02 0.03 0.31 0.29
EN AW-6060 O,
Y0
= 30 N/mm², Eel = 70000 N/mm², Epl = 539.3 N/mm², IY = 52.1 mm4, L = 40 mm
7
Journal of Materials Processing Tech. 276 (2020) 116391
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Fig. 12. Loading and unloading curve of one spring.
loading curve. In fact, its slope is greater, which is due to residual stresses arising during the unloading. 2.2.4. Resulting interference pressure For calculating the interference pressure p between the inner and outer beams, the loading and unloading curves of the inner and outer beams have to be superimposed (compare to Fig. 5). The beams are in contact and thus feature the same midpoint deflection w L 2 . The addition of their specific loads, which mainly depend on the material properties, leads to the sum curve representing the total external load qI + qO . Therefore, at the zero crossing of the sum curve, all external loads become zero. However, plastic deformation causes a remaining joint deflection wsum L 2 = w I L 2 = wO L 2 . The difference in springback then leads to a remaining contact pressure p between the joining partners. This pressure is of compressive nature for the inner part and of expansive nature for the outer part (compare Fig. 4b). To convert the line force N mm into interference pressure N mm2 , the contact surface is taken into account, consisting of the line load length LqI+qO and beam width b.
( )
( )
(
Fig. 13. Local unloading in elastic (c = 0) and elastic-plastic domains (c
0) .
of Table 1, the curve f (x ) is depicted in Fig. 14. It can be seen that the function f (x ) fails to fulfill the boundary conditions at x =
( ) L
Eel Iy w''''constraint (x ) = 0,
(25)
wconstraint (0) = w constraint (0) = 0,
(26)
w'constraint (L /2) = f ' w'''constraint (L /2) = f '''
L = 2
(28)
The solution of this boundary value problem is shown in Eq. (29).
wconstraint (x ) =
x2 L
(
(29)
This leads to the springback function in Eq. (30).
wspringback (x ) = f (x )
wconstraint (x )
(30)
The deflection of the unloaded beam is then given by Eq. (31).
wunloaded (x ) = w loaded (x )
wspringback (x ).
(31)
In particular,
w4 |q = 0 = wunloaded (L /2).
)
The analytically calculated interference pressures p for different external loads (in the form of hydraulic pressure pi ) yield a characteristic curve. Its course, displayed in Fig. 16, shows a linear behavior between four inflection points, with two of them enclosing a process window for joining operations. The inflection points are the result of a change in material behavior caused by the increase in the external load. First, both beams show an elastic behavior. After pressurization and elastic recovery, there is no residual displacement and therefore no interference pressure between the beams. By increasing the pressure, a first yield point at the clamped side of the inner beam occurs. Since yielding creates a residual deflection, the residual interference pressure p can be calculated. This point defines a minimum value for the external pressure load pmin . The
(27)
L =0 2
)
( )
2.3. Deviation of the process window
2 , namely w 2 = 0. This is evident, since the definition of the curve f (x ) is only a differential equation of second order. Therefore, an elastic deflection wconstraint (x ) has to be superimposed to the unconstrained springback, defined through Eq. (25) to Eq. (28). L
( )
(32)
For the example under consideration, the results are illustrated in Fig. 15. By including the point q (w4 ) into the diagram of Fig. 12, it can be seen that the curve of linear unloading is not parallel to the initial
Fig. 14. Exemplary result for the springback deflection f (x ) . 8
Journal of Materials Processing Tech. 276 (2020) 116391
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strength of a force-fit joint. 3. Experimental methodology This section presents the experimental methodology, which can be divided into two parts:
• Joining rectangular profiles by the die-less hydroforming process, • Determining the joint strength in pull-out tests. 3.1. Die-less hydroforming of profiles with rectangular cross-sections Experimental investigations are performed to validate the calculation of the deflection w in the analytical approach. Fig. 17 shows the experimental setup. The axial length aI of the inner joining partner is 280 mm. It has a 50 x 50 mm squared cross-section with a wall thickness tI of 5 mm. The extruded profiles are made of Aluminum EN AW-6060 O. The specimens are heated to 580 °C for one hour and cooled down afterwards at a rate of 100 K per hour to room temperature. The outer joining partner also has a squared cross-section and an axial length aO of 50 mm. Its wall thickness tO is varied from 3 mm to 5 mm resulting in cross-section dimensions of 56 x 56 mm and 60 x 60 mm. The profiles are also extruded from Aluminum EN AW-6060, but in a hardened T6 condition. The initial yield stress Y0 of the annealed inner joining partner is 30 MPa and 250 MPa for the hardened outer joining partner. Prior to joining, the profiles are cleaned in an ultrasonic bath with acetone in order to remove production residues and greases from the contact area. The joining process can be broken down to positioning, hydraulic expansion, and elastic recovery. During positioning, the pressurization region of the hydro probe is symmetrically aligned with respect to the axial center of the outer joining partner. For pressurization, a hydraulic unit from the Maximator company is used. As shown in Fig. 18, the fluid pressure pi is increased with a gradient of 0.5 MPa/s up to the corresponding target value, followed by a pressure holding time of 20 s . A pressure transducer is coupled to a measurement device. For the joining partners, which are assumed in the analytical approach as elastic-plastic springs, a property of interest is the maximum deflection w L 2 . This value is measured by displacement transducers placed at the middle of the outer joining partner (Fig. 17).
Fig. 15. Deflection w (x ) under load and after elastic recovery.
Fig. 16. Analytic-based process window.
second inflection point shows the initiation of a second yield point at the symmetrical end of the inner beam. The final two inflection points occur when the outer beam yields under similar load conditions. At the upper limit, a further increase of external pressure pi does not lead to a significant increase in the internal pressure p anymore. A similar behavior was observed by Marré (2009) for rotationally symmetric joints. Marré defines a physical upper bound as the point where the elastic recovery of the outer joining partner reaches its limit due to the occurance of plastic deformation. He states that this is a second important limit in the form of a maximum value of the external pressure pi,max . Also, it is possible to calculate the interference pressure p for each external load. The interference pressure eventually defines the joint
( )
3.2. Pull-out test The pull-out force FAX is related to the interference pressure p by Eq. (33),
FAX = p A µ
Fig. 17. Joining by die-less hydroforming, experimental setup. 9
(33)
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F. Weber, et al.
measuring system and a sprayed speckle pattern on the samples. A line with an initial length of 50 mm is inserted into the pull-out test pictures from the Aramis system to evaluate the change in length and hence, the strain component in axial direction. To avoid additional deformation in the joining zone, caused by a clamping process in the test setup, a pull-out device is used to support one end of the outer joining partner in the axial direction (see Fig. 19). The pull-out speed is set to a constant velocity of 0.1 mm/s . According to Marré (2009), there can be a subsequent increase in strength after joint failure due to particles released by the pull-out process. These particles are pressed into the surfaces between the joining partners again at further relative movement. Therefore, most of the joints fail before reaching the maximum strength. Hence, the evaluation of the force strain curve is done according to the failure criterion displayed in Fig. 20. The axial connection strength Fpl0.01 is defined as the force, which leads to a relative axial movement between the two joining partners with a plastic elongation of = 0.01%. This is defined as a threshold value. According to Reinhardt (2010), this threshold value corresponds to the technical elasticity limit and thus serves as an indicator for the transition from elastic to plastic joint behavior.
Fig. 18. Time evolution curves of fluid pressure pi and deflection w .
4. Results and discussion Five parameters of the analytical approach can be used for validation. These are:
• Maximum deflection w ( ) (loaded and unloaded) • p and p • Interference pressure p
Fig. 19. Setup of the pull-out-test.
L
i,min
2
i,max
4.1. Profile deflection
( )
Fig. 21 shows the maximum deflection w L 2 of the loaded configuration and Fig. 22 shows the same for the elastically recovered joining partners. Both configurations indicate an increasing maximum deflection w with an increasing specific pressure. The deviation between the profiles, both experimentally and analytically, is caused by the change in the second moment of area Iy . With regard to Fig. 11, an increase of Iy results in a decreasing deflection. Up to the value of the analytically determined value of pi,max , it can be observed that the analytical model overestimates the increase in deflection compared to the experimental data. It is found out that the deviation between analytically and experimentally determined deflection is a result of an inaccurate flow curve linearization (see Fig. 23). With regard to Eq. (7) this leads to an underestimation of the modulus Eelpl and an overestimation of the deflection w , since Eelpl is a demoninator in the equation for w . A modification of Epl results in a larger modulus Eelpl and thus a smaller deflection w (x ) .
Fig. 20. Failure criterion for determination of pull-out force FAX .
where A is the contact surface and µ the Coulomb friction coefficient. To determine the joint strength in the form of the axial pull-out-force, the joined specimens are quasi-statically separated. The tests are conducted in a Zwick Z250 universal testing machine, which measures the pull-out force. The strain measurement is done optically via Digital Image Correlation (DIC) with an Aramis 4 M (from the GOM Company)
Fig. 21. Comparison of analytical and experimental results of the deflection wmax in the loaded condition. 10
Journal of Materials Processing Tech. 276 (2020) 116391
F. Weber, et al.
Fig. 22. Comparison of analytical and experimental results of the deflection wmax after elastic recovery (unloaded).
Fig. 23. Improved flow curve linearization.
Fig. 24. Comparison of analytical and experimental results regarding the deflection wmax in the loaded condition with adapted Eelpl .
Fig. 25. Comparison of analytical and experimental results regarding the deflection wmax after elastic recovery with adapted Eelpl .
The flow curve is linearized until the end of uniform elongation. To get a better estimate of Epl in the actual process, numerical simulations are performed in Abaqus with the above mentioned materials and
geometries for a discontinuous pressure application, following the procedure of Müller et al. (2016). The joining partners are modeled with solid brick elements and the mesh size is determined by a 11
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Table 2 Decrease in deviation of analytics from experiments of w of the adapted flow curve linearization.
( ) at p L
2
i,max
one wall thickness with minimal deviation between the analytically and experimantlly determined pull-out force. The analysis shows a good agreement for an adhesion coefficient v lr of 0.17 for a wall thickness of 5 mm and 0.32 for a thickness of 3 mm. The difference in the adhesion coefficients of the two wall thicknesses (3 mm → lr = 0.32, 5 mm → lr = 0.17) can be explained by a superimposed bending moment in axial direction, which is neglected in the one-dimensional analytical approach. The axial bending moment is a result of a difference between the actual length of the pressurization area and the length of the outer joining partner. A larger wall thickness leads to a larger restoring moment in the thick-walled outer joining partner. After elastic recovery, this increases the pressure p in the contact area and thus leads to a lower adhesion coefficient lr . It can also be observed that the gradient of the specific interference pressure increasement as a function of the specific hydraulic pressure is about 1.12 and 1.35 (tO= 5 mm and tO= 3 mm respectivly) until pi,max . After this upper limit, the gradients decrease to 0.065 and 0.056 for tO= 5 mm and tO= 3 mm respectivly. This proves that a higher pressure does not lead to a significant increase in joint strength after exceeding pi,max . The maximum deviation of the analytically calculated maximum pressure pi,max is 11.6% (tO= 3 mm) and 10.8% (tO= 5 mm) compared to the experiments. Thus, the crucial process parameter pi,max can be predicted with sufficient accuracy.
as a result
tO
Under load
After elastic recovery
3 mm 5 mm
48.6 % 72.7 %
44.0 % 61.5 %
5. Conclusion The joining by die-less hydroforming process is extended to profiles with rectangular cross-sections. A new analytical approach is developed to calculate the main process parameters required for process design. The main results are as follows:
Fig. 26. Comparison of analytical and experimental results regarding the specific interference pressure pi Y0 .
• The pressurization range is required for the selection of an appro-
sensitivity analysis. Taking into account the results of this modeling, the equivalent strain ¯ of 0.02 at pi,max is much lower than assumed so far. Linearizing the flow curve until this value increases Epl because of a higher initial gradient of the flow curve, as shown in the adapted values in Fig. 23. Fig. 24 shows the displacement under load and Fig. 25 shows the displacement after elastic recovery by taking into account such an adaption of Eelpl . It leads to an improved accuracy of the model, as displayed in Table 2. The values for pi,min and pi,max do not change significantly through this modification because the equation for the yield moment MY (Eq. 18) contains the sum of the moduli Eel and Epl in its denominator, and since Eel > > Epl , there is no significant difference at the beginning of plastic deformation.
•
4.2. Pressurization range and interference pressure Fig. 26 contains the results of the pull-out tests and the analytical calculation for both of the investigated wall thicknesses of the outer joining partner. The plot shows the specific connection strength p Y0 plotted against the corresponding specific fluid pressure pi Y0 . The initial yield strength Y0 of the inner joining partner is taken into account because it is the weaker one and fails at first in a tensile test. In the experiments, the pull-out force FAX was measured and evaluated to obtain Fpl0.01. For determining the experimental interference pressure p , Eq. (34) can be used.
Fpl0.01 = p A v lr
•
(34)
Instead of using the the Coulomb friction value µ , which sets the experimentally determined pull-out force in relation to the experimentally determined normal force, the adhesion coefficient v lr is used, as is permissable according to DIN 7190. The adhesion coefficient represents the ratio of experimentally determined pull-out force and analytically calculated interference pressure p and thus also includes some inherent assumptions, such as a constant contact area A . Using the method of least squares, v lr can be found for a measuring series of
priate hydraulic unit and is a significant parameter for process monitoring. It can be predicted analytically by calculating the pressure limits pi,min and pi,max as in the proposed model. The lower limit pi,min represents the pressure when plastic deformation of the inner joining partner occurs first. At the more significant upper limit pi,max , plastic deformation of the outer joining partner starts. At this point, the achievable joint strength will not increase significantly since the limit of elastic springback is reached. The experimentally determined deflection w (x ) , that is also used to validate the analytical approach, deviates from the analytical values. This can be explained by an overestimation of the equivalent reference strain ¯ needed for the flow curve linearziation. By taking into account the actual equivalent strain (determined by numerical investigations), the deviation between the analytical and the experimental data at pi,max can be reduced by 48.6% for the profiles with a wall thickness of 3 mm under load and by 44.0% after elastic recovery. In case of the profiles with 5 mm wall thickness, the deviation is reduced by 72.7% under load and 61.5% after elastic recovery. Further improvement of the flow curve linearization procedure, for example a piecewise linearization, can reduce the remaining deviation between analytics and experiments. With the analytical approach, it is also possible to calculate the remaining interference pressure p after elastic recovery, which is essential for predicting the joint strength. This value is correlated to experimental results. The model still has to be improved for a more accurate prediction of the achievable joint strength. However, it is possible to reduce future experimental efforts due to the capability of the model to determine the upper pressurization limit pi,max .
Acknowledgments This work is based on results from project TE508/50-1, which is kindly supported by the German Research Foundation (DFG). The authors thank Ms. Bhuvi Nirudhoddi for her efforts in proofreading. 12
F. Weber, et al.
Journal of Materials Processing Tech. 276 (2020) 116391
References
Berücksichtigung von Naben mit nichtkonstantem Außendurchmesser. Dr.-Ing.Dissertation. TU Clausthal. Jantscha, R., 1929. Über das Einwalzen und Einpressen von Kessel- und Überhitzerrohren bei Verwendung verschiedener Werkstoffe. Dr.-Ing.-Dissertation. TH Darmstadt. Liu, F., Zhenga, J., Xua, P., Xub, M., Zhu, G., 2004. Forming mechanism of double-layered tubes by internal hydraulic expansion. Int. J. Press. Vessel. Pip. 81 (7), 625–633. Marré, M., 2009. Grundlagen der Prozessgestaltung für das Fügen durch Weiten mit Innenhochdruck. Dr.-Ing.-Dissertation. Technische Universität Dortmund. Marré, M., Andreas, R., Heuser, R., Tekkaya, A. E., 2013. Verfahren zum lokalen Fügen und/oder zum lokalen Umformen von Hohlprofilen mittels Hochdruck. Patent DE 10 2010 012 452 B8. Mori, K., Bay, N., Fratini, L., Fabrizio, M., Tekkaya, A.E., 2013. Joining by plastic deformation. CIRP Ann. Manuf. Technol. 62, 673–694. Müller, M., Gies, S., Tekkaya, A.E., 2016. Gesenkfreies Innenhochdruckfügen von Vierkantrohren. 7. VDI-Fachtagung Welle-Nabe-Verbindungen 2016, 09.-10.11.2016, Karlsruhe, Germany. pp. 241–246. Müller, M., Gies, S., Tekkaya, A.E., 2018. Joining by die-less hydroforming for profiles with oval cross section. Key Eng. Mater. 767, 405–412. Osweiller, F., 1988. French rules for the design of fixed-tubesheet heat exchangers. Int. J. Press. Vessel. Pip. 32 (5), 335–354. Podhorsky, M., Krips, H., 1990. Wärmetauscher – Aktuelle Probleme der Konstruktion und Berechnung. Vulkan-Verlag, Essen ISBN: 3-8027-2296-5. Reinhardt, H.-W., 2010. Ingenieurbaustoffe 2. vollst. überarbeitete Auflage. Ernst & Sohn Verlag, Berlin. Yokell, S., 1992. Expanded and welded-and-expanded tube-to-Tubesheet joints. J. Press. Vessel Technol. 114, 157–165.
Barnes, T.A., Pashby, I.R., 2000. Joining techniques for aluminium spaceframes used in automobiles: part I and II. J. Mater. Process. Technol. 99 (1–3), 62–79. Both, J., Brueggemann, T., Dosch, S., Elser, J., Kronthaler, M., Morasch, A., Otter, M., Pietzka, D., Selvaggio, A., Weddeling, C., Wehrle, E., Wirth, F.X., Baier, H., Biermann, D., Brosius, A., Fleischer, J., Lanza, G., Schulze, V., Tekkaya, A.E., Zabel, A., Zaeh, M.F., 2011. Advanced manufacturing and design techniques for lightweight structures. Aluminium 9-2011, 60–64. DIN 7190-1:2017-02, 2017. Pressverbände – Teil 1: Berechnungsgrundlagen und Gestaltungsregeln für zylindrische Pressverbände. DIN Deutsches Institut für Normung e. V., Beuth Verlag GmbH. DIN EN ISO 6892-1:2017-02, 2017. Metallische Werkstoffe – Zugversuch – Teil 1: Prüfverfahrenbei Raumtemperatur. DIN Deutsches Institut für Normung e. V., Beuth Verlag GmbH. Garzke, M., 2001. Auslegung innenhochdruckgefügter Pressverbindungen unter Drehmomentbelastung. Dr.-Ing.-Dissertation. Technische Universität Clausthal. Gies, S., Weddeling, C., Marré, M., Kwiatkowski, L., Tekkaya, A.E., 2012. Analytic prediction of the process parameters for form-fit joining by die-less hydroforming. Key Eng. Mater. 504-506 (2012), 393–398. Gies, S., Weddeling, C., Kwiatkowski, L., Tekkaya, A.E., 2013. Groove filling characteristics and strength of form-fit joints produced by die-less hydroforming. Key Eng. Mater. 554-557 (2013), 671–680. Grünendick, T., 2004. Die Berechnung innenhochdruckgefügter Pressverbindungen. Dr.Ing.-Dissertation. TU Clausthal Düsseldorf. Halle, M., 2012. Die Berechnung innenhochdruckgefügter Pressverbindungen unter
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Contents lists available at ScienceDirect
Journal of Materials Processing Tech. journal homepage: www.elsevier.com/locate/jmatprotec
Analytical process design for interference-fit joining of rectangular profiles F. Weber , M. Müller, P. Haupt, S. Gies, M. Hahn, A.E. Tekkaya ⁎
T
Institute of Forming Technology and Lightweight Components, TU Dortmund University, Baroper Straße 303, 44227, Dortmund, Germany
ARTICLE INFO
ABSTRACT
Associate Editor: Z Cui
The classical joining of round tubes by die-less hydroforming is extended to joining of rectangular profiles. A new analytical approach for the calculation of required process parameters is developed. It allows the calculation of the interference pressure p and the required fluid pressure range given by a lower and upper limit ( pi,min and pi,max ). These limits are the result of the deformation state of the inner and outer joining partner. The minimum pressure, where plastic deformation of the inner joining partner occurs first, characterizes the value at which the joint formation starts. The maximum pressure is the value, at which plastic deformation of the outer joining partner occurs. The analytical model is validated by experimental studies where two rectangular aluminum profiles (EN AW-6060) with different heat treatment conditions were joined. Above pi,max , the increase of the achievable joint strength with respect to the internal pressure is not significant anymore. The maximum deviation between the analytically and experimentally determined maximum fluid pressure pi,max is about 12% in case of a wall thickness of 3 mm and 11% in case of a wall thickness of 5 mm.
Keywords: Joining by forming Die-less hydroforming Intereference fit Profiles with rectangular cross-sectional shapes Analytical prediction
1. Introduction In the automotive industry, lightweight strategies like spaceframe structures are used (Barnes and Pashby, 2000) for well-known reasons. Lightweighting is often realized by the use of lightweight materials. The combination of two different materials leads to multi-material design. This strategy requires new joining technologies to create multi-material parts. In recent years it has been shown that joining by die-less hydroforming (DHF) is a suitable method for joining components with circular cross-sections such as camshafts (Garzke, 2001) and multi-layer pipes (Liu et al., 2004). The main advantage besides the joining of dissimilar materials is the elimination of additional joining elements, e.g. bolts, rivets or screws (Mori et al., 2013). Simple process management, the possibility of easy monitoring and the lack of heat input motivate the application of this joining technology. As depicted in Fig. 1a, the joint can be achieved by either force-fit, form-fit as well as a superposition of both mechanisms. According to Garzke (2001), the process can be divided into three main steps as displayed in Fig. 1b. In the first step, the joining partners, consisting of the outer and inner tube, are aligned in a shaft to hub configuration and the joining tool is placed inside the inner tube, underneath the joining zone. After that, a pressurized fluid (water or hydraulic oil) is applied into the gap between the tool and the inner tube. The seals limit the axial length of the pressurization area. As the pressure increases, the joining partners
⁎
begin to expand. Plastic deformation of the inner tube occurs when the inner stress reaches the tube’s initial yield strength Y0 (Yokell, 1992). A relief of the fluid pressure leads to the elastic recovery of both joining partners. The irreversible deformation of the inner tube prevents the full springback of the elastically expanded outer ring, which leads to a remaining interference pressure in the contact area. The interference pressure will increase with higher fluid pressure until an upper limit is reached. At this limit, a further increase in the fluid pressure leads to no significant increase in joint strength since the outer ring is plastically deformed and the maximum springback is reached (Marré, 2009). The strength of the joint can futher be increased by form-fit joining. In formfit joining, the material bulges into the provided grooves on the inside of the ring, which leads to an undercut (Podhorsky and Krips, 1990). First mentioned by Jantscha (1929); Podhorsky and Krips (1990) brought the process into industrial application for the production of tube heat exchangers. For this so-called tube-to-tube sheet joints, designing rules were given by Osweiller (1988). For his investigations, he created pipe to pipe-plate joints by an internal high pressure. There are investigations and physical models, describing the bulging process of the inner tube into the groove (Gies et al., 2012) to calculate the working fluid pressure and the influence of the elastic springback in the case of form-fit joining (Gies et al., 2013). Garzke (2001) investigated the behavior of joints under torsional loads and developed a physical model for the calculation of an average interference pressure. The model was combined with an empirical approach to increase its
Corresponding author. E-mail address: [email protected] (F. Weber).
https://doi.org/10.1016/j.jmatprotec.2019.116391 Received 10 January 2019; Received in revised form 10 August 2019; Accepted 9 September 2019 Available online 10 September 2019 0924-0136/ © 2019 Elsevier B.V. All rights reserved.
Journal of Materials Processing Tech. 276 (2020) 116391
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Nomenclature
ML Elastic limit moment MY Yield moment p Inferference pressure pi , Fluid pressure pi,min , pi,max Minimum / maximum fluid pressure pi,target Experimental target fluid pressure Q Lateral force of the inner / outer joining partner q Line load of the inner / outer joining partner t Wall-thickness of the inner / outer joining partner Elastic stress across the beam’s cross-section xx,el Elastic-plastic stress across the beam´s cross-section xx,elpl Yield stress Y Initial yield stress Y0 µ Coulumb friction coefficient v lr Adhesion coefficient w Deflection of the beam wconstraint Elastic deformation constraint wdeformed Remaining deflection after holding time w loaded wunloaded Deflection in loaded / unloaded condition wmax Deflection at target fluid pressure wspringback Deflection due to springback wsum Remaining deflection y Coordinate in direction of profile length x Coordinate in direction of profile width x1, x2 Coordinate of transition from elastic to elastic-plastic domain xq Initial coordinate from where the partial line load q is acting z Coordinate in direction of profile height Profile height where elastic-plastic deformation occurs zY
Symbol/meaning1
A a b c cL c unloaded cY D Eel Epl Eelpl ¯ xx
el pl
F Fpl0.01 FAX Feq f h Iy L Li LqI+qO lf M Melpl
Contact surface Axial length of the inner / outer joining partner Width of the beam´s cross-section Curvature Elastic limit curvature Unloading curvature Yield curvature Diameter of the O-ring Young´s modulus of the inner / outer joining partner Plastic modulus of the inner / outer joining partner Elastic-plastic modulus Equivalent reference strain Strain in the beam´s cross-section Elastic elongation Plastic elongation Spring force of the inner / outer joining partner Axial connection strength Pull-out force Total spring force Deflection of local unloading Height of the beam´s cross-section Moment of inertia Length of the beam’s cross-section Length of the inner side of the inner / outer joining partner Length of the partial line load qI + qO , Length of the joining zone Bending moment of the inner / outer joining partner Moment in elastic-plastic area
accuracy in case of thick-walled outer joining partners. Grünendick (2004) published a model for the calculation of the interference pressure, taking into account the deformation in the radial and axial direction and investigated the connection behavior for oscillating loads. Marré (2009) developed an analytical approach for interference fit joining, in which he derived equations to calculate the required process variables such as minimum and maximum fluid pressure. Halle’s (2012) approach also allows calculations for interference fit joints. The difference to the other approaches is that the outer joining partner is thickwalled with a rotationally symmetric joining zone and a non-rotationally symmetric mass allocation. An adapted mass allocation can be found, for example, in lightweight optimized drive components. An extension of the process limits in terms of the geometry of the joining partners is given by Müller et al. (2018) for joining by hydraulic expansion of joining partners with oval cross-sectional shapes. Both et al. (2011) proved in experimental investigations that the process is also suitable for rectangular profiles. In their investigations, they used a tool as shown in Fig. 2, which was developed and patented by Marré et al. (2013). The process principle is identical to the joining of rotationally symmetric components whereas the tool design is different. The prototype tool has two seal concepts for the joining zones, a discontinous and a continuous one around the profile. Discontinuous joining zones (see Fig. 2a) are created by the O-ring-seals on each outer surface of the hydro probe where the pressurized fluid is applied. Continuous joining zones (see Fig. 2b) are made via two circumferential grooves for elastomer seals. Müller et al. (2016) published results of experimental and numerical investigations of both concepts. They state that, by assuming elastic behavior
of the outer joining partner, there is no significant difference between the two joining zone concepts regarding the internal stress distribution. The numerical results showed the same springback characteristic for both concepts. Therefore, the remaining interference pressure, which correlates with the achievable joint strength, is nearly the same. For joining rectangular profiles (see Fig. 3a and c), existing physical models cannot be used because they implicate a different springback behavior which is only valid for axisymmetric tubes where the springback is a result of the circumferential stress. Since springback in rectangular profiles occurs because of the bending stresses, a new approach has to be developed to determine pi,min , pi,max and p . Fig. 3b displays the loading characteristic with the main process parameters, including the deflection w of the profiles in the loaded and unloaded condition. Based on this loading situation, an analytical model is derived and presented in the next section. Section three describes the experimental methodology for the corresponding validation. The analytical and experimental results are then compared in section four. 2. Analytical model 2.1. Model assumptions A model is developed for the joining of profiles with squared crosssections employing the discontinuous sealing concept (Fig. 2a). Therefore, the profile’s cross-section is divided into four parts. Each side is considered as a clamped beam under a partial line load qI + qO , which represents the sum of the specific loads of the inner and outer beam, as shown in Fig. 4a. The length of the inner and outer beam is LI,i and LO,i respectively. The length of the line load in Eq. (1) is defined as the O-rings diameter.
1 In this arcticle the index I is used for the inner joining partner whereas O is used for variables concerning the outer joining partner.
LqI+qO = D 2
(1)
Journal of Materials Processing Tech. 276 (2020) 116391
F. Weber, et al.
Fig. 1. a) Joining mechanisms and b) process sequence for joining by die-less hydroforming (DHF) (Garzke, 2001).
Fig. 2. Prototype tool concepts (Marré, 2013) a) Discontinuous joining zone b) Continuous joining zone.
Due to the symmetry of the load and the boundary conditions of the clamped sides (w = w = 0 ), the analysis of one half of a beam is sufficient. The beams, representing the inner and outer joining partner, are in contact. This will be incorporated by considering them as elasticlinear plastic springs in parallel connection coupled by the equal deflection w of their midpoints as given in Eq. (2) (compare Fig. 5a).
wI
L L = wO 2 2
2.2. Extension of the Euler-Bernoulli beam theory 2.2.1. Elastic and elastic-plastic bending behavior For purely linear-elastic beams, the bending moment M (x ) , as a function of the curvature c (x ) = w (x ) , the moment of inertia Iy and the Young´s modulus Eel are related by Eq. (3).
M (x ) = (2)
Eel Iy c (x )
(3)
The product Eel Iy is called elastic bending stiffness. In the Euler-Bernoulli bending theory of elastic beams, it is assumed that the strain = xx and accordingly the stress = xx = Eel xx is linearly distributed over the cross-section (Fig. 6). The deformation of the beam under a line force q (x ) is represented by the deflection w (x ) , which can be calculated through integration of the governing differential Eq. (4) by taking into account known boundary conditions.
A linear elastic spring force is calculated as the product of the spring constant and the deflection. The total force Feq in parallel connection is the sum of the spring forces. For the two beams it is the sum of the line loads, qI and qO. In case of pure elasticity, both springs unload to zero if the load Feq is reduced to zero. However, for an elastic-linear plastic behavior of the springs, it is possible that an internal force arises during unloading. This is qualitatively sketched in Fig. 5b. If the load qI + qO on the two beams in contact is reduced to zero, an internal pressure p between them can be observed. For a quantitative analysis of this situation, the Euler-Bernoulli beam theory has to be extended to elastoplasticity.
Eel Iy w '''' (x ) = q (x )
(4)
The strain within a material fiber is proportional to the distance z from the neutral fiber and the curvature c (x ) , as seen in Eq. (5). 3
Journal of Materials Processing Tech. 276 (2020) 116391
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Fig. 3. a) Application case: Squared profiles b) Aims of the analytical approach for interference fit c) Lightweight frame structure of the Collborative Research Center SFB TR10.
=
xx (x ,
z) =
c z=
(5)
z w''(x )
The displayed elastic-plastic equations derive similar to the elastic equation but with the elastic plastic modulus Eelpl and by moving the equation by the value of ±zY along the z -axis and ± Y0 along the xx -axis. Integration of the bilinear function yields the bending moment as a function of curvature. Observing the symmetry of the stress distribution, the elastic-plastic bending moment is given by Eq. (9).
The stress distribution for linear elasticity is then calculated by Eq. (6). xx (x ,
z) =
Eel z w '' (x ) =
Eel z c
(6)
In order to extend these ideas to elastoplasticity, the assumption of a linear strain distribution is retained; but the stress distribution is replaced by the bilinear approximation for linear isotropic hardening with an elastic-plastic modulus in Eq. (7), where Eel = el is the stress-strain ratio in the elastic region and Epl = is the ratio in the pl plastic region.
Eelpl
=
Eel
+
Epl
E
= elpl
Melpl = 2 b
Y0
Eel c
z dz =
zY 0
xx (z )
z dz +
h /2 zY
xx (z )
z dz. (9)
(7)
Melpl (c ) =
The elastic and elastic-plastic stress distribution is illustrated in Fig. 7 for a negative value of the curvature. With the initial flow stress Y0 , the coordinate limiting the elastic region is given by Eq. (8).
zY =
xx (z )
For negative values of c the result of the calculation is given by Eq. (10).
Eel Epl Eel + Epl
h /2 0
Eel Eel + Epl
3
Iy h
Y
b
3 1 Y 3 Eel 2 c 2
Epl Iy c
(10)
Substituion of Iy for the given cross-section in Fig. 6 yields Eq. (11).
Melpl (c ) =
(8)
Eel b Eel + Epl
Y
4
h2
3 1 Y 3 Eel 2 c 2
Epl
h3 c 12
For positive values of c the same calculation yields Eq. (12).
Fig. 4. a) Model assumptions regarding the geometrical aspects b) Free body diagram. 4
(11)
Journal of Materials Processing Tech. 276 (2020) 116391
F. Weber, et al.
Fig. 5. a) Analogous spring model b) Schematic loading-unloading curve of elastic-plastic springs.
Melpl (c ) =
Eel Eel + Epl
3
Iy
Y
h
3 1 Y 3 Eel2 c 2
+b
Melpl ( ± c ) =
Epl Iy c
(12)
h h , i. e. 2 2
MY =
Y0
(13)
Eel |c|
2 Y0 Eel h
cY = (14)
b h2 6
Y0
.
Melpl (c ) =
Eel Eel + Epl
Iy
Melpl (c ) =
Eel Eel + Epl
=
3
Iy h
cL b
Y
h 3
c
Y
3
1 3 Eel 2 c 2 Y
+b
Y
3
3 Eel
2
1 c2
Epl Iy c
for c
cY ]
[c Y , [
(17)
(18)
MY 3 Y = (Eel Iy ) (Eel + Epl ) h
(19)
cY
c
cY cY cY
(20)
2.2.2. Loading cases and material states The material state of a beam depends on the bending moment caused by the external load. In Fig. 10, different possible states of loading of the left half of the beam are shown qualitatively. For each state, an external load q is applied from x q to L 2 . Initially, the beam deforms elastically for an increasing load, as seen in Fig. 10a, until the bending moment M (x = 0) approaches the value MY . As the external load increases, an elastic-plastic deformation arises at the clamping point. Thus, an expanding region of elastic-plastic behavior occurs within the area between x = 0 and x = x1, (see Fig. 10b). This is valid until a second elastic-plastic area arises starting from the midpoint x = L 2 . A further increase of the loading q leads to a second
Epl Iy c for c
c L. (16) The function M (c ) is depicted in Fig. 8 (solid line). There is a smooth transition from elastic to elastic-plastic states. For the further procedure, it is convenient to replace this function by its asymptotic approximation (Fig. 8, dashed line). This can be done by 3
,
(compare Fig. 9) The applied approach leads to a bilinear approximation of the moment-curvature relation. This facilitates an integration of the beam´s deflection curve.
M (c ) cL
Y
M (c ) MY Eelpl l y (c + cY ) if c MY Eelpl l y (c c Y ) if c
Considering this, a sectionally defined function M (c ) is given by Eq. (16).
Eel Iy c for
3 Iy Eel (Eel + Epl ) h
Eel l y c if
(15)
Mel (c ) =
]
The value of ML indicates plastsicization at the outer fiber of the beam, whereas the beam is fully plastic at MY . The elastic-plastic bending properties are then represented by Eq. (20).
Substituing Eq. (14) into Eqs. (10) and (12) respectively leads to the elastic limit moment ML in Eq. (15).
ML (c L ) =
c Y )for c
and
This gives the elastic limit curvature cL according to Eq. (14).
cL =
Eelpl Iy (c
with
The elastic description of the beam is valid for
|zY|
MY
simply neglecting the nonlinear term Y 2 2 for Melpl (c ) in Eq. (16). 3 Eel c This leads to the new definition in Eq. (17), 1
Fig. 6. a) Cross-section of the beam b) Linear stress distribution over the beams cross-section. 5
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Fig. 7. Elastic and elastic plastic stress distribution over the beams cross-section (for c < 0).
Each general solution depends on four integration constants. The 16 } ) can be deconstants (Cj, Ej, Dj and Fj for {j | 1 j 4 j termined from the boundary conditions at the beam ends and the continuity requirements for the deflection w , the slope w = , the bending moment M and the lateral force Q at the transition points. 2.2.3. Resulting load-deflection relationship The objective of the following analysis is to establish a characteristic relation between the intensity of the external load q and the deflection
( )
of the beam’s midpoint w L 2 . Therefore, the boundary value problem given in Fig. 11 has to be solved in four steps:
1.) Elastic solution for the whole beam (Fig. 10a). Increase of the load q from 0 to the value of the yield moment MY at x = 0. This defines the first part of the characteristic curve q (w1) . This part is linear in contrast to the following parts. 2.) Solution of the situation in Fig. 10b. The determination of the transition point x = x1 between elastic-plastic and elastic behavior is part of the problem (root of a polynomial). As the load q is raised incrementally, x1 increases step by step, whereas the bending moment M L 2 increases simultaneously up to MY . This determines a point q (w2) of the characteristic curve. 3.) Solution of the situation of Fig. 10c. Now there are two transition points x1 and x2 to be calculated from M1 (x1, x2) = MY and M2 (x1, x2) = MY (roots of two polynomials depending on two variables). This calculation is repeated with increasing values of q, and stopped before x1 becomes larger than x q . This defines the last point q (w3) of the characteristic curve. 4.) Solution for unloading ((w4)= 0). This is addressed in more detail hereafter.
Fig. 8. Function of the bending moment M (c) .
( )
The characteristic curve for the loading and unloading of one spring (parameter values in Table 1) is sketched in Fig. 12. The listed material parameters ( Y0 , Eel , Epl ) of the inner beam in Table 1, as well as the below mentioned parameters of the outer beam, are determined by uniaxial tensile tests as laid down in DIN EN ISO 6892. Note that the shape of the curve between w1 and w3 is only indicated by straight lines, whereas it is nonlinear in reality. However, the values q (w1) , q (w2) and q (w3) are exact. The distribution of the bending moment is different from zero in the unloaded configuration q (w4 )= 0, because the support of the beam is statically undetermined. In order to get this distribution (and springback deflection w4 with q (w4)= 0), the principle of local unloading is employed, i.e. the bending moment is reduced at every point to zero. This produces a change in curvature, c unloaded (x ) , as visualized in general in Fig. 13. This change in curvature is the second derivative f (x ) of a function f (x ) , which can be interpreted to be the springback deflection of the unloaded beam. This curvature reduces to zero within the elastic
Fig. 9. Elastic-plastic bending properties as a function of curvature.
area with elastic-plastic material behavior from x2 to L 2 (see Fig. 10c). For each situation, a separate differential equation has to be solved. In case of purely elastic bending (Fig. 10a), the beam has to be divided into two areas. The area near to the clamped side is unloaded, followed by a loaded zone. The case of Fig. 10b is divided into three domains. At the clamped side, there is an unloaded elastic-plastic region (small bending stiffness), followed by a second unloaded but elastic region (large bending stiffness). The remaining region is also just elastic but loaded. The procedure for solving the differential equation is exemplarily shown in detail for the situation shown in Fig. 10c. One symmetric half of the beam is divided into four domains. As seen in Fig. 11, each domain is characterized by its load case (loaded/unloaded) and its material state, represented by a bending stiffness (elastic/elastic-plastic). 6
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Fig. 10. Load cases: (a) Pure elastic bending (b) Elastic-plastic bending (c) Elastic-plastic bending in two domains.
Fig. 11. Differential equations for a beam with elastic-plastic domains.
regions, whereas in the plastic regions it follows from the elastic recovery. Thus, the springback is defined as follows in Eq. (21) to Eq. (24) for the regions 1 to 4:
c unloaded,1 =
M1 (x ) = f1 ''(x ) (Eel Iy )
c unloaded,3 = 0 = f3 ''(x ) c unloaded,4 =
(21)
(24)
This defines the curve f (x ) by integration. In doing so, the initial conditions f (0) = f (0)= 0 have to be considered as well as the continuity requirement of f and f at the transition points. For the example
(22)
c unloaded,2 = 0 = f2 ''(x )
M4 (x ) = f4 ''(x ) (Eel Iy )
(23)
Table 1 Parameter values. Bending situation
q [N/mm]
w I [mm]
Material
Elastic (Fig. 11a): q(w1) Elastic-plastic (Fig. 11b): q(w2) Elastic-plastic (Fig. 11c): q(w3) Unloading q (w4 )
26.24 33.05 51.13 0
0.02 0.03 0.31 0.29
EN AW-6060 O,
Y0
= 30 N/mm², Eel = 70000 N/mm², Epl = 539.3 N/mm², IY = 52.1 mm4, L = 40 mm
7
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Fig. 12. Loading and unloading curve of one spring.
loading curve. In fact, its slope is greater, which is due to residual stresses arising during the unloading. 2.2.4. Resulting interference pressure For calculating the interference pressure p between the inner and outer beams, the loading and unloading curves of the inner and outer beams have to be superimposed (compare to Fig. 5). The beams are in contact and thus feature the same midpoint deflection w L 2 . The addition of their specific loads, which mainly depend on the material properties, leads to the sum curve representing the total external load qI + qO . Therefore, at the zero crossing of the sum curve, all external loads become zero. However, plastic deformation causes a remaining joint deflection wsum L 2 = w I L 2 = wO L 2 . The difference in springback then leads to a remaining contact pressure p between the joining partners. This pressure is of compressive nature for the inner part and of expansive nature for the outer part (compare Fig. 4b). To convert the line force N mm into interference pressure N mm2 , the contact surface is taken into account, consisting of the line load length LqI+qO and beam width b.
( )
( )
(
Fig. 13. Local unloading in elastic (c = 0) and elastic-plastic domains (c
0) .
of Table 1, the curve f (x ) is depicted in Fig. 14. It can be seen that the function f (x ) fails to fulfill the boundary conditions at x =
( ) L
Eel Iy w''''constraint (x ) = 0,
(25)
wconstraint (0) = w constraint (0) = 0,
(26)
w'constraint (L /2) = f ' w'''constraint (L /2) = f '''
L = 2
(28)
The solution of this boundary value problem is shown in Eq. (29).
wconstraint (x ) =
x2 L
(
(29)
This leads to the springback function in Eq. (30).
wspringback (x ) = f (x )
wconstraint (x )
(30)
The deflection of the unloaded beam is then given by Eq. (31).
wunloaded (x ) = w loaded (x )
wspringback (x ).
(31)
In particular,
w4 |q = 0 = wunloaded (L /2).
)
The analytically calculated interference pressures p for different external loads (in the form of hydraulic pressure pi ) yield a characteristic curve. Its course, displayed in Fig. 16, shows a linear behavior between four inflection points, with two of them enclosing a process window for joining operations. The inflection points are the result of a change in material behavior caused by the increase in the external load. First, both beams show an elastic behavior. After pressurization and elastic recovery, there is no residual displacement and therefore no interference pressure between the beams. By increasing the pressure, a first yield point at the clamped side of the inner beam occurs. Since yielding creates a residual deflection, the residual interference pressure p can be calculated. This point defines a minimum value for the external pressure load pmin . The
(27)
L =0 2
)
( )
2.3. Deviation of the process window
2 , namely w 2 = 0. This is evident, since the definition of the curve f (x ) is only a differential equation of second order. Therefore, an elastic deflection wconstraint (x ) has to be superimposed to the unconstrained springback, defined through Eq. (25) to Eq. (28). L
( )
(32)
For the example under consideration, the results are illustrated in Fig. 15. By including the point q (w4 ) into the diagram of Fig. 12, it can be seen that the curve of linear unloading is not parallel to the initial
Fig. 14. Exemplary result for the springback deflection f (x ) . 8
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strength of a force-fit joint. 3. Experimental methodology This section presents the experimental methodology, which can be divided into two parts:
• Joining rectangular profiles by the die-less hydroforming process, • Determining the joint strength in pull-out tests. 3.1. Die-less hydroforming of profiles with rectangular cross-sections Experimental investigations are performed to validate the calculation of the deflection w in the analytical approach. Fig. 17 shows the experimental setup. The axial length aI of the inner joining partner is 280 mm. It has a 50 x 50 mm squared cross-section with a wall thickness tI of 5 mm. The extruded profiles are made of Aluminum EN AW-6060 O. The specimens are heated to 580 °C for one hour and cooled down afterwards at a rate of 100 K per hour to room temperature. The outer joining partner also has a squared cross-section and an axial length aO of 50 mm. Its wall thickness tO is varied from 3 mm to 5 mm resulting in cross-section dimensions of 56 x 56 mm and 60 x 60 mm. The profiles are also extruded from Aluminum EN AW-6060, but in a hardened T6 condition. The initial yield stress Y0 of the annealed inner joining partner is 30 MPa and 250 MPa for the hardened outer joining partner. Prior to joining, the profiles are cleaned in an ultrasonic bath with acetone in order to remove production residues and greases from the contact area. The joining process can be broken down to positioning, hydraulic expansion, and elastic recovery. During positioning, the pressurization region of the hydro probe is symmetrically aligned with respect to the axial center of the outer joining partner. For pressurization, a hydraulic unit from the Maximator company is used. As shown in Fig. 18, the fluid pressure pi is increased with a gradient of 0.5 MPa/s up to the corresponding target value, followed by a pressure holding time of 20 s . A pressure transducer is coupled to a measurement device. For the joining partners, which are assumed in the analytical approach as elastic-plastic springs, a property of interest is the maximum deflection w L 2 . This value is measured by displacement transducers placed at the middle of the outer joining partner (Fig. 17).
Fig. 15. Deflection w (x ) under load and after elastic recovery.
Fig. 16. Analytic-based process window.
second inflection point shows the initiation of a second yield point at the symmetrical end of the inner beam. The final two inflection points occur when the outer beam yields under similar load conditions. At the upper limit, a further increase of external pressure pi does not lead to a significant increase in the internal pressure p anymore. A similar behavior was observed by Marré (2009) for rotationally symmetric joints. Marré defines a physical upper bound as the point where the elastic recovery of the outer joining partner reaches its limit due to the occurance of plastic deformation. He states that this is a second important limit in the form of a maximum value of the external pressure pi,max . Also, it is possible to calculate the interference pressure p for each external load. The interference pressure eventually defines the joint
( )
3.2. Pull-out test The pull-out force FAX is related to the interference pressure p by Eq. (33),
FAX = p A µ
Fig. 17. Joining by die-less hydroforming, experimental setup. 9
(33)
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measuring system and a sprayed speckle pattern on the samples. A line with an initial length of 50 mm is inserted into the pull-out test pictures from the Aramis system to evaluate the change in length and hence, the strain component in axial direction. To avoid additional deformation in the joining zone, caused by a clamping process in the test setup, a pull-out device is used to support one end of the outer joining partner in the axial direction (see Fig. 19). The pull-out speed is set to a constant velocity of 0.1 mm/s . According to Marré (2009), there can be a subsequent increase in strength after joint failure due to particles released by the pull-out process. These particles are pressed into the surfaces between the joining partners again at further relative movement. Therefore, most of the joints fail before reaching the maximum strength. Hence, the evaluation of the force strain curve is done according to the failure criterion displayed in Fig. 20. The axial connection strength Fpl0.01 is defined as the force, which leads to a relative axial movement between the two joining partners with a plastic elongation of = 0.01%. This is defined as a threshold value. According to Reinhardt (2010), this threshold value corresponds to the technical elasticity limit and thus serves as an indicator for the transition from elastic to plastic joint behavior.
Fig. 18. Time evolution curves of fluid pressure pi and deflection w .
4. Results and discussion Five parameters of the analytical approach can be used for validation. These are:
• Maximum deflection w ( ) (loaded and unloaded) • p and p • Interference pressure p
Fig. 19. Setup of the pull-out-test.
L
i,min
2
i,max
4.1. Profile deflection
( )
Fig. 21 shows the maximum deflection w L 2 of the loaded configuration and Fig. 22 shows the same for the elastically recovered joining partners. Both configurations indicate an increasing maximum deflection w with an increasing specific pressure. The deviation between the profiles, both experimentally and analytically, is caused by the change in the second moment of area Iy . With regard to Fig. 11, an increase of Iy results in a decreasing deflection. Up to the value of the analytically determined value of pi,max , it can be observed that the analytical model overestimates the increase in deflection compared to the experimental data. It is found out that the deviation between analytically and experimentally determined deflection is a result of an inaccurate flow curve linearization (see Fig. 23). With regard to Eq. (7) this leads to an underestimation of the modulus Eelpl and an overestimation of the deflection w , since Eelpl is a demoninator in the equation for w . A modification of Epl results in a larger modulus Eelpl and thus a smaller deflection w (x ) .
Fig. 20. Failure criterion for determination of pull-out force FAX .
where A is the contact surface and µ the Coulomb friction coefficient. To determine the joint strength in the form of the axial pull-out-force, the joined specimens are quasi-statically separated. The tests are conducted in a Zwick Z250 universal testing machine, which measures the pull-out force. The strain measurement is done optically via Digital Image Correlation (DIC) with an Aramis 4 M (from the GOM Company)
Fig. 21. Comparison of analytical and experimental results of the deflection wmax in the loaded condition. 10
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Fig. 22. Comparison of analytical and experimental results of the deflection wmax after elastic recovery (unloaded).
Fig. 23. Improved flow curve linearization.
Fig. 24. Comparison of analytical and experimental results regarding the deflection wmax in the loaded condition with adapted Eelpl .
Fig. 25. Comparison of analytical and experimental results regarding the deflection wmax after elastic recovery with adapted Eelpl .
The flow curve is linearized until the end of uniform elongation. To get a better estimate of Epl in the actual process, numerical simulations are performed in Abaqus with the above mentioned materials and
geometries for a discontinuous pressure application, following the procedure of Müller et al. (2016). The joining partners are modeled with solid brick elements and the mesh size is determined by a 11
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Table 2 Decrease in deviation of analytics from experiments of w of the adapted flow curve linearization.
( ) at p L
2
i,max
one wall thickness with minimal deviation between the analytically and experimantlly determined pull-out force. The analysis shows a good agreement for an adhesion coefficient v lr of 0.17 for a wall thickness of 5 mm and 0.32 for a thickness of 3 mm. The difference in the adhesion coefficients of the two wall thicknesses (3 mm → lr = 0.32, 5 mm → lr = 0.17) can be explained by a superimposed bending moment in axial direction, which is neglected in the one-dimensional analytical approach. The axial bending moment is a result of a difference between the actual length of the pressurization area and the length of the outer joining partner. A larger wall thickness leads to a larger restoring moment in the thick-walled outer joining partner. After elastic recovery, this increases the pressure p in the contact area and thus leads to a lower adhesion coefficient lr . It can also be observed that the gradient of the specific interference pressure increasement as a function of the specific hydraulic pressure is about 1.12 and 1.35 (tO= 5 mm and tO= 3 mm respectivly) until pi,max . After this upper limit, the gradients decrease to 0.065 and 0.056 for tO= 5 mm and tO= 3 mm respectivly. This proves that a higher pressure does not lead to a significant increase in joint strength after exceeding pi,max . The maximum deviation of the analytically calculated maximum pressure pi,max is 11.6% (tO= 3 mm) and 10.8% (tO= 5 mm) compared to the experiments. Thus, the crucial process parameter pi,max can be predicted with sufficient accuracy.
as a result
tO
Under load
After elastic recovery
3 mm 5 mm
48.6 % 72.7 %
44.0 % 61.5 %
5. Conclusion The joining by die-less hydroforming process is extended to profiles with rectangular cross-sections. A new analytical approach is developed to calculate the main process parameters required for process design. The main results are as follows:
Fig. 26. Comparison of analytical and experimental results regarding the specific interference pressure pi Y0 .
• The pressurization range is required for the selection of an appro-
sensitivity analysis. Taking into account the results of this modeling, the equivalent strain ¯ of 0.02 at pi,max is much lower than assumed so far. Linearizing the flow curve until this value increases Epl because of a higher initial gradient of the flow curve, as shown in the adapted values in Fig. 23. Fig. 24 shows the displacement under load and Fig. 25 shows the displacement after elastic recovery by taking into account such an adaption of Eelpl . It leads to an improved accuracy of the model, as displayed in Table 2. The values for pi,min and pi,max do not change significantly through this modification because the equation for the yield moment MY (Eq. 18) contains the sum of the moduli Eel and Epl in its denominator, and since Eel > > Epl , there is no significant difference at the beginning of plastic deformation.
•
4.2. Pressurization range and interference pressure Fig. 26 contains the results of the pull-out tests and the analytical calculation for both of the investigated wall thicknesses of the outer joining partner. The plot shows the specific connection strength p Y0 plotted against the corresponding specific fluid pressure pi Y0 . The initial yield strength Y0 of the inner joining partner is taken into account because it is the weaker one and fails at first in a tensile test. In the experiments, the pull-out force FAX was measured and evaluated to obtain Fpl0.01. For determining the experimental interference pressure p , Eq. (34) can be used.
Fpl0.01 = p A v lr
•
(34)
Instead of using the the Coulomb friction value µ , which sets the experimentally determined pull-out force in relation to the experimentally determined normal force, the adhesion coefficient v lr is used, as is permissable according to DIN 7190. The adhesion coefficient represents the ratio of experimentally determined pull-out force and analytically calculated interference pressure p and thus also includes some inherent assumptions, such as a constant contact area A . Using the method of least squares, v lr can be found for a measuring series of
priate hydraulic unit and is a significant parameter for process monitoring. It can be predicted analytically by calculating the pressure limits pi,min and pi,max as in the proposed model. The lower limit pi,min represents the pressure when plastic deformation of the inner joining partner occurs first. At the more significant upper limit pi,max , plastic deformation of the outer joining partner starts. At this point, the achievable joint strength will not increase significantly since the limit of elastic springback is reached. The experimentally determined deflection w (x ) , that is also used to validate the analytical approach, deviates from the analytical values. This can be explained by an overestimation of the equivalent reference strain ¯ needed for the flow curve linearziation. By taking into account the actual equivalent strain (determined by numerical investigations), the deviation between the analytical and the experimental data at pi,max can be reduced by 48.6% for the profiles with a wall thickness of 3 mm under load and by 44.0% after elastic recovery. In case of the profiles with 5 mm wall thickness, the deviation is reduced by 72.7% under load and 61.5% after elastic recovery. Further improvement of the flow curve linearization procedure, for example a piecewise linearization, can reduce the remaining deviation between analytics and experiments. With the analytical approach, it is also possible to calculate the remaining interference pressure p after elastic recovery, which is essential for predicting the joint strength. This value is correlated to experimental results. The model still has to be improved for a more accurate prediction of the achievable joint strength. However, it is possible to reduce future experimental efforts due to the capability of the model to determine the upper pressurization limit pi,max .
Acknowledgments This work is based on results from project TE508/50-1, which is kindly supported by the German Research Foundation (DFG). The authors thank Ms. Bhuvi Nirudhoddi for her efforts in proofreading. 12
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