- Email: [email protected]
Optimization methods for the tube hydroforming process applied to advanced high-strength steels with experimental verification
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
journal homepage: www.elsevier.com/locate/jmatprotec
Optimization methods for the tube hydroforming process applied to advanced high-strength steels with experimental verification Nader Abedrabbo a,∗ , Michael Worswick a , Robert Mayer b , Isadora van Riemsdijk c a
Department of Mechanical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada b General Motors Research & Development Center, Warren, MI 48090, United States c Dofasco Inc. (ArcelorMittal) Research & Development, Hamilton, ON L8N 3J5, Canada
a r t i c l e
i n f o
a b s t r a c t
Article history:
In this paper, an optimization method linked with the finite element method is presented for
Received 20 September 2007
developing forming parameters of the tube hydroforming (THF) process for several advanced
Received in revised form
high-strength steel (AHSS) materials. The goal of this research was to maximize formability
17 January 2008
by identifying the optimal internal hydraulic pressure and end-feed rate, while satisfying
Accepted 19 January 2008
the failure limits defined by the forming limit diagram (FLD). The optimization software HEEDS was used in combination with the nonlinear structural finite element code LS-DYNA to carry out the investigation. The pressure and feed profiles identified through the auto-
Keywords:
mated optimization procedures were obtained with accuracy and with no trial-and-error
Optimization
experiments. Experimental validation of the optimized profiles by hydroforming multiple
Tube hydroforming
tubes made from different materials was then performed successfully. © 2008 Elsevier B.V. All rights reserved.
HEEDS LS-DYNA Forming limit diagrams High-strength steels Formability
1.
Introduction
Weight reduction is a key priority for improving automotive fuel economy. Multiple candidates for replacing mild steel in automotive structures have been proposed, e.g. advanced high-strength steels (AHSS), aluminum, magnesium, and composites. Advanced high-strength steels, in particular, are attractive candidate materials, offering higher strength for energy absorption and the opportunity to reduce weight through use of thinner gauges. However, though steels in general are highly formable, the increase in strength achieved by
∗
the AHSS materials is associated with a partial compromise in formability. In recent years, tube hydroforming (THF) technology has become a popular method, especially in the automotive industry, for producing complex three-dimensional structural shapes because of its enormous advantages over the more traditional processes. There are several advantages to hydroforming over conventional processes such as stamping and welding. These advantages include part consolidation, weight reduction due to improved part design, improved structural strength and stiffness, and reduction in the associated
Corresponding author. Tel.: +1 519 885 1211x35887; fax: +1 519 885 5862. E-mail address: [email protected] (N. Abedrabbo). 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.01.060
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
tooling and material costs (Dohmann and Hartl, 1996). In view of these advantages, the range of parts currently being produced or developed using THF by the automotive industry continues to grow. These parts include engine cradles, radiator supports, roof side rails, exhaust manifold assemblies, camshafts, sub-frames and instrument support panels (Ferrier, 1996; Ni, 1994). Two methods are widely used to hydroform circular tubes: low-expansion (low-pressure) and high-expansion (high-pressure) hydroforming processes. A key difference between the two processes is that the low-expansion process introduces only limited circumferential expansion with no end-feeding required to form the tube. In the high-expansion method, the circumferential strains are large, causing excessive tube thinning to occur and therefore end-feeding of the tube, i.e. extra tube material pushed into the die, is used to counter the thinning problem. Tubes formed using the lowexpansion method are generally easy to manufacture, with very little failure or wrinkling problems occurring (or none at all). The success of the high-expansion process, on the other hand, is highly dependent on a number of variables, including material formability, friction conditions, and most importantly, the loading paths (pressure and end-feeding) used to form the tube. Gao et al. (2002) classified the tube hydroforming process based on loading parameters and concluded that the pressure dominant process to be the most challenging load curve determination problem since it involves a high risk for wrinkling, bursting and leakage. Therefore, parameters which are derived from this process would be the most beneficial for load curve optimization (Jansson et al., 2007). The loading paths in the THF process are traditionally determined using trial-and-error procedures and, in particular, rely on past experiences of the operator (Fann and Hsiao, 2003). The THF process is further complicated if new materials and die geometries are used for which the operator has no prior knowledge. The traditional process is both timeconsuming and expensive. Furthermore, there is no guarantee that “optimum” pressure and end-feed profiles can be found by the trial-and-error process. Therefore, in order to determine the pressure and end-feed hydroforming profiles for any material and die geometry, an integrated approach to the problem comprised of the finite element analysis (FEA) of the hydroforming process, a failure model, and an optimization code is required. Numerical analysis is a critical tool for understanding the complex deformation mechanics that occur during sheet forming processes. Finite element analysis and simulations are used in automotive design and formability processes to predict deformation behaviour accurately during stamping operations. Finite element method (FEM) simulation of the hydroforming process has been proven to be an advantageous tool in assisting automotive designs. Ni (1994) and Wu and Yu (1996) simulated engine cradle components. Over the years, several research papers have been published about the optimization of the tube hydroforming process. In these papers, multiple optimization techniques have been proposed. Johnson et al. (2004) proposed a method in which the optimization algorithm uses the material stress–strain curve and the deformation theory of plasticity to relate the current stress and strain increments to the
111
next applied load increments. In this method, a controlled increment in plastic strain was prescribed for the next solution increment, and the pressure and end-feed increments were calculated to give a constant ratio of incremental axial and hoop strains. Imaninejad et al. (2005) used LS-DYNA and the optimization software LS-OPT to optimize the internal pressure and axial feed of a T-joint design in which a minimum thickness variation was chosen as a design objective and maintaining the effective stress below the materials ultimate stress. Fann and Hsiao (2003) used the conjugate gradient optimization methods to optimize a T-shaped tube using LS-DYNA. Yingyot et al. (2004) investigated two methods; self-feeding and adaptive simulation techniques, in order to optimize the tube hydroforming process and experimental versus numerical results were presented. Manabe et al. (2006) studied the optimization process of tube hydroforming using a virtual-database-assisted fuzzy control system to determine the forming parameters. Also, Jansson et al. (2007) proposed an adaptive optimization method based on the use of the response surface methods. In the current research, the optimization software Hierarchical Evolutionary Engineering Design System (HEEDS), which uses genetic algorithm (GA) search methods and utilizes a unique search strategy called Simultaneous Hybrid Exploration that is Robust, Progressive and Adaptive (SHERPA), which is an innovative method that uses multiple search methods simultaneously (not sequentially), was used in combination with the nonlinear structural finite element code LS-DYNA (Hallquist, 2006) to optimize the process parameters and determine the best loading paths for the tube hydroforming process.
2.
THF experiments
2.1.
Experimental setup
Tube hydroforming of circular tubes was performed using square-shaped dies with two different corner radii inserts: 6 and 18 mm. Tubes were hydroformed using a 1000-tonnes (9806 kN) macrodyne hydroforming press at the University of Waterloo Labs, shown in Fig. 1. Fig. 2 shows a sectional view of the high-expansion die used to hydroform the tubes with the 18 mm corner radius insert attached. Fig. 3 shows a cross-section schematic of the 6 mm die insert used in the research with die dimensions. The sloped area shown in Fig. 3 provides a smooth transition between the round tube section and the final square die shape. To illustrate the differences between the two types of hydroforming processes usually used in THF, Fig. 4 shows a comparison between the low- and high-expansion tubes at the end of the hydroforming process. Also shown in the graph is the original tube size used in the two types of experiments before hydroforming. In the low-expansion hydroforming process, the initial tube size is larger than the die. As the die closes, it mechanically forms the tube to the die shape with a small fluid pressure present to prevent buckling. After the die is closed, the fluid pressure is gradually increased to form the tube. In the high-expansion hydroforming process, the tube is initially in contact with the sides of the die, as shown. As fluid
112
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 3 – Cross-section of the 6 mm insert showing dimensions (mm).
Fig. 1 – The macrodyne 1000 tonnes (9806 kN) forming press at the University of Waterloo.
In order for the tubes to be hydroformed using the highexpansion process, i.e. to counter tube thinning due to the higher expansion, end-feeding was introduced using two endfeed actuators, each with 1112 kN (250 kip) force at each end of the tube. End-feeding pushes material into the die, thereby increasing the formability of the tubes. One of the end-feed actuators used in the high-expansion experiments is shown in Fig. 5. In order to seal the tube during hydroforming, the tip of the end-feed actuators, which incorporates a Teflon Oring and a high-pressure polymeric seal, is inserted inside the tube a distance of 70 mm where the edge of the tube sits on a shoulder. The tip of the actuator used in the experiments showing the sealing O-rings are shown in Fig. 6.
2.2.
Experimental procedure
Tubes were cut to a length of 785 mm and the ends of the tubes were deburred to prevent damage to the Teflon and polymeric O-rings. All tube specimens were lubricated using a solid-film lubricant, HydroDraw 625, which was first sprayed onto the tube and allowed to dry. Twist-compression tests with the die steel (P20) and sheet steels of the tube materials were performed. The Coulomb’s coefficient of friction was determined
Fig. 2 – High-expansion hydroforming top and bottom dies with the 18 mm insert attached.
pressure is increased, tube expansion occurs to fill the corner region. In this case, no mechanical forming occurs before fluid pressure is applied. As seen in Fig. 4, the high-expansion process causes higher expansion of the tube (between 18 and 26% change in perimeter depending on corner radius) compared to the low-expansion process (approximately 0.8% change in perimeter).
Fig. 4 – Comparison between the original round tube size (NH) and the low- and high-expansion hydroforming processes for the 6 mm corner radius dies. The small square is the final low-expansion tube (LP), while the larger square is for the high-expansion (HP) process tube.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 5 – One of the 1112 kN (250 kip) end-feed actuators used in the high-expansion hydroforming experiments.
Fig. 6 – Tip of the end-feed actuator showing the Teflon ring and polymeric O-ring used to seal the tube. The tube rests on the shoulder during the hydroforming process.
to be approximately 0.05, which was used in the numerical analysis. After placing the tubes in the die cavity and closing the dies, the end-feed actuators were advanced to allow the tube ends to rest on the shoulders of the end-feeds. Air was removed using a vacuum pump, and the tube was then filled with water. Next, the loading paths, i.e. end-feed displacement versus time and fluid pressure versus time, were applied to hydroform the tube. The LabView program was used to control the process variables. In order to ensure accurate pressure application for every end-feed displacement point, a pressure versus displacement profile was used to control the process. If the system response time was slow, or in case of intermittent pressure drop, the system stopped the end-feed actuators from advancing until the pressure reaches the required level, after which the hydroforming process continues. This interactive procedure prevented the tube from either wrinkling (due to excessive feed) or tearing (due to high-pressure).
2.3.
113
Fig. 7 – A sample of a tube formed using the high-expansion process with non-optimized forming profiles using the 6 mm die. Excessive thinning caused the tube to burst before it formed to the die shape.
tube to untimely burst. In this case, the amount of extra tube material pushed into the die to counter the thinning effects was not enough to prevent tube bursting. Fig. 7 shows a sample of a tube hydroformed using the high-expansion process with a minimum amount of end-feeding for which tube burst occurred. As the pressure in the tube increased, the tube began to expand from the middle of the tube, starting at the regions of the tube in contact with die (see Fig. 4) and continuing into the unsupported regions at the corners, causing the material to thin. Expansion continued until the excessive thinning in the material caused the tube to burst before the tube formed to the die’s final shape. In the second case, to overcome the excessive thinning problem, a certain amount of end-feeding of the tube was used. However, the amount of end-feeding required for each tube depended on the geometry of the die (6 or 18 mm corner radius), friction conditions, and the type of material being hydroformed. Therefore, end-feeding of each tube material was determined such that enough material was fed into the die to prevent excessive thinning. It is important to note, however, that material overfeeding will cause the longitudinal compressive stress to be higher than the hoop tensile stress in the material, causing undesirable buckling or wrinkling in the tube. Fig. 8 depicts the case of overfeeding. Although the tube was formed without bursting, the extra material pushed into the die caused buckling in the tube along the corner radius where the material is unsupported and free to buckle.
THF using non-optimized forming profiles
To illustrate the importance of optimizing the forming profiles, two cases are shown next where non-optimized hydroforming profiles were used to form the tubes. In the first case, the tubes were formed with a minimal amount of end-feeding. Due to a combination of expansion and friction between the die and the tube surfaces, tube thinning occurred, causing the
Fig. 8 – A tube formed using the high-expansion hydroforming process with excessive amount of end-feeding. The extra material pushed in caused the tube to buckle as shown by the area in the dashed line.
114
3.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Materials
In the current research, monolithic tubes made from multiple materials and wall thicknesses were considered. Materials used include the following: conventional deep drawing quality (DDQ) steel of wall thickness 1.8 mm; high-strength low alloy (HSLA-350) steel with wall thicknesses of 1.5 and 1.8 mm; and AHSS materials comprising the dual phase (DP) alloys DP600 with wall thickness of 1.8 mm and DP780 with wall thickness of 1.5 mm. High-strength low alloy and DP steels are currently being used to reduce the weight of automobiles (Gorni and Mei, 2005). Not only do they display good formability and weldability, but also they benefit from the high-strength of martensite with reduced alloying and without the costly inclusion of postfabrication heat treatment. All tubes in this research had an outer diameter of 76.2 mm.
3.1.
Constitutive equations (flow stress)
Finite element analysis and simulations are vital tools in understanding the deformation behaviour that occurs during forming and crash experimental operations. Confidence in the numerical analysis depends on the use of a constitutive model that accurately describes the behaviour of the material being investigated. Therefore, in this research, full material characterization was performed on the different materials used in order to extract the required parameters of the constitutive models to be used in the numerical analysis. The materials were tested in both the sheet and the as-formed tube conditions in order to identify tube forming operations’ effects on tube properties. For the as-formed tube material, since rolling operations used to manufacture the tubes induce varying levels of workhardening around the circumference of the tube, testing was performed on specimens from three positions on the perimeter of the tube, corresponding to the 3, 6, and 9 O’clock positions (the weld seam is located at 12 O’clock) as shown in Fig. 9. From these experimental tensile tests, the parameters of the Power-Law constitutive model (flow stress) were fit to the experimental results for each material. The Power-Law model
is represented as follows: (¯ ¯ εp ) = K(¯εp + ε0 )
n
(1)
where “K” (strength-hardening coefficient) and “n” (strainhardening exponent) are material constants. ε¯ p is the effective plastic strain and ε0 is a constant representing the elastic strain to yield. Although multiple tensile tests were performed for each material in different directions (sheet data, 3, 6, and 9 O’clock tube positions), and the constitutive parameters were calculated for each direction, only one set of parameters is used in the numerical analysis. In order to represent the tube in its multiple directions in the numerical analysis, a weighted average method was used to calculate the Power-Law constitutive parameters. The following formula was used: X avg =
2(X3 O clock + X6 O clock ) 3
(2)
where X represents a general variable. This specific averaging method was used because tensile test results indicated that the work-hardening that is imparted to the tube during the tube manufacturing process is symmetric but not uniform. Therefore, the 3 and 9 O’clock positions were found to be very close and thus considered equal. Table 1 lists the constitutive parameters of the Power-Law model for all the materials used in this research.
3.2.
Failure criteria
In order to determine the required hydroforming profiles using the combination of the optimization program and the FEA analysis, a failure criteria for the different materials must be determined. Including the failure criteria in the FEA program helps to identify both the time and the location of failure in the numerical model. In the current project, failure prediction was based on the stress-based forming limit diagrams (-FLD). In forming processes, if the loading path is sufficiently close to linear (proportional loading) then a general strain-based FLD can be used to assess failure of the part. However, for a general forming process in which the loading path may not be linear (non-proportional loading), it becomes necessary to determine the strain-based FLD for each element based on its loading path. This process is computationally expensive, and therefore, a path-independent failure method should be used instead. Stoughton (2001) proposed a method to transform the path-dependent strain-based FLD into a pathindependent stress-based FLD. A review of different types of
Table 1 – Constitutive parameters for the different tubular materials used
Fig. 9 – Top view of cross-section of tube showing locations of test samples used in tensile test and also showing the weld seam location.
Material
t0 (mm)
OD (mm)
K (MPa)
n
ε0
DDQ HSLA-350 HSLA-350 DP-600 DP-780
1.8 1.5 1.8 1.8 1.5
76.2 76.2 76.2 76.2 76.2
578.14 684.00 679.88 900.00 1166.37
0.183 0.095 0.121 0.109 0.130
7.46E-4 1.81E-3 1.49E-3 2.24E-3 2.60E-3
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
115
and satisfying the stress-based forming limit diagram (-FLD). The optimization problem can be defined as follows: Maximize:
Displacement of node “n” (Fig. 11)
Subject to:
All elements of tube model to lie below -FLD
By varying:
Fig. 10 – Stress-based forming limit diagrams (-FLDs) for all materials.
-FLD and their use in FEA can be found in Stoughton and Zhu (2004). Fig. 10 shows the -FLD for all the materials used in this research which were determined from the -FLD’s as described by Stoughton (2001).
4.
THF process optimization
4.1.
Problem definition
Pressure:
Pmin < Pi < Pmax
End-feed force:
Fmin < Fi < Fmax , where i:1–10 (depending on curve profile segments)
Pmax and Fmax represent the maximum pressure and end-feed force possible in the current hydroforming press, respectively. The profiles were setup to increase in a monotonic fashion. In the current setup, since the end-feed actuators used in the experiments have a maximum force of 1112 kN (250 kip), the best method to ensure that the results obtained from the numerical simulations satisfy this limitation was to use the end-feed force as an optimization variable instead of the end-feed displacement. After the optimum solutions were found, the end-feed displacements to be used experimentally were then easily determined from the numerical end-feed displacement. The acquired profiles were then used in the experimental setup to successfully form all tubes (a minimum of six tubes were formed for each material).
4.2.
The optimization objective in the current research was to determine the corresponding internal fluid pressure schedule and the end-feed displacement that would allow the tube to expand to maximum extent without bursting or buckling. By considering a point “n” on the corner of the tube, shown in Fig. 11, as the tube expands into the die, the distance “U” traveled by the point “n” increases. The optimal rates and values of axial feed and internal pressure are determined so as to maximize the distance ‘U’ while avoiding severe thinning
Fig. 11 – Schematic of the optimization process. “n” is a center node, “P” is hydroforming pressure, and “U” is node displacement.
HEEDS optimization program
The optimization software HEEDS was used in combination with the nonlinear structural finite element code LS-DYNA to carry out the numerical optimization. The characteristics of the design space associated with the current optimization problem were not known a priori. In this case, it was advisable to employ a combination of global and local search techniques in order to achieve a broad and effective search for an optimal solution. For such problems, HEEDS utilizes a combination of evolutionary, gradient-based, and design-ofexperiments search heuristics (HEEDS, 2008). By intelligently coupling global and local search techniques, the HEEDS optimization algorithms are able to find excellent solutions to even the most challenging design problems. Local optimization methods (e.g., nonlinear sequential programming, response surface methods, etc.) are valuable for fine-tuning a design, but not for exploring different design concepts in an effort to identify a much better design. Because the mathematical cost or objective functions associated with many practical design problems are multi-modal (i.e., they have many peaks and valleys) or even discontinuous, the use of global search methods (e.g., parallel genetic algorithms) improves the likelihood of achieving significant design innovation. While global methods search broadly over a large design space, local optimization methods simultaneously focus on promising sub-regions of the design space to identify the best designs in that region. HEEDS applies several optimization methods simultaneously, allowing each method to take advantage of the best attributes and solutions found from other parallel searches. The multiple semi-independent search processes exchange
116
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
information about the solution space with each other, helping to jointly satisfy multiple constraints and objectives. This search method is called a heterogeneous multi-agent approach. This approach quickly identifies design attributes with good potential and uses them to focus, improve and accelerate the search for an optimum solution. More details about the HEEDS program can be found at www.redcedartech.com.
4.2.1.
Genetic algorithm
HEEDS employs a GA to perform evolutionary search. GAs are particularly useful when the design space is large and complex. The main problem with using a simple GA is the potentially large number of design evaluations required to obtain a set of satisfactory solutions. HEEDS reduces the number of evaluations required to obtain a set of satisfactory solutions by hierarchically decomposing a problem with multiple agents that represent the problem in various ways, while combining efficient local search methods (e.g., response surface methods, nonlinear sequential quadratic programming, and simulated annealing). Genetic algorithm is a search procedure that is based on the mechanics of natural selection. Specific knowledge is embedded in a chromosome (or design vector) which represents a possible design with a set of values of all the design variables. The number of choices per design variable determines the fidelity (or resolution) of each design variable. These design variables are the building blocks used to construct a particular design. The GA creates and destroys designs during a process that involves decoding each chromosome, evaluating its satisfaction of constraints and its performance relative to the objectives then allowing a simulated “natural selection” to determine which designs are eliminated and which survive to generate other derivative designs. Designs that perform well (relative to constraints and objectives) have a higher probability of surviving to influence future designs (their “offspring”). During reproduction, the two genetic operators commonly modeled that produce new chromosomes (or design vectors) are called crossover and mutation.
4.2.2.
Crossover
The crossover operation (sometimes also called “recombination”) forms a new solution by combining parts of two existing solutions. A high crossover rate (fraction of population replaced by crossover during one generation of reproduction) will produce many new designs in each generation, but will also have a high probability of disrupting (and potentially losing, at least temporarily) higher-performance designs already found, and requires more evaluations of constraints and objectives in each generation which are typically the most costly computing operation in the entire problem.
4.2.3.
resemble each other, thereby making it difficult for crossover to generate solutions that differ very much from the current set). A set of co-existing designs defines a population, while successive populations are termed generations. That is, each period during which a set of existing solutions are evaluated then used with natural selection, crossover, and mutation to generate a new set of solutions, is called a generation. A large population typically contains more genetic diversity (i.e., different values of design variables) that typically improves the ultimate results of the GA search. However, the more new individuals created in each generation, the more computer time must be spent evaluating the constraints and objectives of the new individuals, so there is a tradeoff that must be made. Within each agent, a GA search begins by creating a single initial population, wherein chromosomes (vectors of design variable values) are randomly created. At this point the performance (constraint satisfaction) and objective values of each design is evaluated. Biased by the evaluations obtained, the GA uses unary (mutation) and binary (crossover) operators on these designs to create another population. This population probabilistically maintains the previously high performing designs while discarding poorly performing designs. New population members are evaluated and then additional rounds of generation and selection are performed. This process is repeated until satisfactory solution(s) are obtained. Incorporating these processes in a computer routine produces an algorithm that solves problems in a manner reminiscent of natural evolution. Independent GA searches in several agents can share information with each other through a user-defined migration process.
4.3.
FE numerical model
The explicit part of the finite element program LS-DYNA was used in the current research. To take advantage of process symmetry, one-eighth of the tube is considered in the FEM optimization model to reduce the simulation time, as shown in Fig. 12. The die and end-feeds were created using rigid shell elements of 2.0 mm × 2.0 mm size with material properties of steel. The tube material was created using 2.0 mm × 2.0 mm Belytschso-Tsay (LS-DYNA Manual, 2007) shell elements.
Mutation
Mutation is a reproduction operation that produces a new solution from a single existing solution through any of several ways. Mutation can change the value of one design variable or of many simultaneously and can change them in uniform random ways, or by a normal distribution (for example around the current values of the design variables). Mutation helps to maintain diversity and reduces the possibility of premature convergence (the tendency of a set of solutions to come closely
Fig. 12 – One-eighth FEM model of the tube hydroforming process used in the optimization process.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 13 – Process flow schematic of the optimization process showing the interaction between HEEDS and LS-DYNA.
117
Fig. 16 – Experimental results for DP600-T1.8 mm tube formed with the high-expansion process and the 6 mm corner radius die using the optimized forming profiles in Fig. 14. Figure shows tube after a trimming process at a length of 400 mm.
Fig. 17 – Experimental results for DP780-T1.5 mm tube formed with the high-expansion process and the 18 mm corner radius die using the optimized forming profiles in Fig. 15. Figure shows tube after a trimming process at a length of 400 mm.
Fig. 14 – Optimized hydroforming profiles for all materials hydroformed using the 6 mm high-expansion die.
Fig. 18 – Experimental results for HSLA350-T1.8 mm tube formed with the high-expansion process and the 6 mm corner radius die using the optimized forming profiles in Fig. 14. Figure shows tube after a trimming process at a length of 400 mm.
Fig. 15 – Optimized hydroforming profiles for all materials hydroformed using the 18 mm high-expansion die.
The user material subroutine (UMAT) option in LS-DYNA was used in this research where the von-Mises constitutive material model was implemented using the cutting-plane algorithm (Simo and Ortiz, 1985; Abedrabbo et al., 2006), which is based on the incremental theory of plasticity. The UMAT procedure was used in order to include the failure criteria into the FEM analysis. The stress-based forming limit curves (-FLD), as shown in Fig. 10, were represented in mathematical form using curve-fitting methods. During the simulation process, each element of the tube was compared
118
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 19 – Predicted and measured engineering strains for DP600-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die.
against the -FLD at every integration point to check whether it failed or not. If failure of an element occurred, an output file was written that contained the displacement of a node “n” on the corner of the tube (see Fig. 11). This output file was then used by HEEDS in the calculation of the next optimized step. The process continued until a suitable solution was reached and the tube was completely formed, i.e. node “n” contacted die at corner, and no element in the tube had failed. A fullscale model of the tube and dies was also implemented in order to compare the numerical results to the experimental ones.
4.4.
HEEDS–LS-DYNA interface
HEEDS is a fully automated optimization program that is modular, generic, and independent. The program can be linked to
any type of solver (commercial or otherwise). HEEDS uses the input file(s) and output file(s) from a baseline (initial) design to identify and analyze design candidates and generate project responses. A basic analysis definition in HEEDS consists of (Abedrabbo et al., 2004):
• A command that executes the solver to be used for the analysis of the design (LS-DYNA in this case). • All input files containing the optimization process variables that are varied for each evaluation (fluid pressure and endfeed force in this project). • At least one output file containing the responses of interest from a successful analysis run which are used to evaluate the performance of the design (node displacement in this case).
Fig. 20 – Predicted and measured strains for DP780-T1.5 mm tube formed using the high-expansion hydroforming process with the 18 mm corner radius die.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
119
Fig. 21 – Predicted and measured engineering strains for HSLA350-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die.
The optimization process and the interaction between the optimization program HEEDS and the FEA code LS-DYNA are represented in the flow schematic of Fig. 13.
in the 6 mm die as compared to the 18 mm die, and therefore a higher level of end-feed is required to prevent the tube from bursting.
4.5.
5. Experimental verification and numerical simulations
Optimization results
Fig. 14 shows the pressure versus end-feed displacement profiles acquired from the optimization process for all the tubes hydroformed using the 6 mm corner radius high-expansion die. Fig. 15 shows the optimization results for all the tubes hydroformed using the 18 mm corner radius high-expansion die. As seen from these two figures, the 6 mm corner radius die requires a higher amount of end-feeding to form the tubes without bursting as compared to the 18 mm corner radius die. This is anticipated since a higher level of tube thinning occurs
5.1.
Experimental validation
The fluid pressure and end-feed profiles determined for each material using the optimization methods described earlier were successfully used to hydroform tubes for all the different materials (as shown in Table 1) using the dies with corner radii of 6 and 18 mm. In total, there were 10 high-expansion numerical simulations performed; 5 for the 6 mm corner radius die
Fig. 22 – Measured and predicted tube wall thickness for DP600-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die.
120
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 23 – Measured and predicted tube wall thickness for DP780-T1.5 mm tube formed using the high-expansion hydroforming process with the 18 mm corner radius die.
and 5 for the 18 mm corner radius die. Experimentally, at least 6 tubes were hydroformed for each corresponding numerical simulation, for a total of 60 tubes. Fig. 16 shows a sample experimental result, after a trimming process, for the DP600-T1.8 mm tube formed with the high-expansion process and the 6 mm corner radius die using the optimized forming profile shown in Fig. 14 for the same material. Fig. 17 shows a sample experimental result, after a trimming process, for the DP780-T1.5 mm tube formed with the high-expansion process and the 18 mm corner radius die using the optimized forming profile shown in Fig. 15 for the DP780 material. Fig. 18 shows another sample experimental result, after a trimming process, for the HSLA350-T1.8 mm tube formed with the high-expansion process and the 6 mm corner radius die using the optimized forming profile shown in Fig. 14 for the same material. All the tubes in this research were experimentally hydroformed without any defects and
with no trial-and-error similar to the three samples shown in Figs. 16–18. The engineering strains within the formed tubes were measured by comparing changes in the shape of circular grids that were etched onto the tube prior to the tube hydroforming operation. These measurements were then compared to values obtained from numerical simulations. All strain measurements were taken at the half-length of the tube. Fig. 19 shows the predicted (numerical) and measured (experimental) engineering strains for the DP600-T1.8 mm tube formed using the high-expansion process with a corner radius of 6 mm. The engineering strains are plotted versus angle around the tube, where the weld seam was at approximately 0◦ . Experimental measurements were made for every other circle grid around the perimeter of the tube. Fig. 20 shows the predicted and measured results for the DP780-T1.5 mm tube hydroformed using the high-expansion
Fig. 24 – Measured and predicted tube wall thickness for HSLA350-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
121
Table 2 – Predicted maximum percentage of thickness change for all materials hydroformed using the high-expansion die with a corner radius of 6 mm Material DDQ-T1.8 mm HSLA350-T1.5 mm HSLA350-T1.8 mm DP600-T1.8 mm DP780-T1.5 mm
Maximum (%) thickness reduction 10.0 11.4 11.7 10.0 9.7
process with a corner radius of 18 mm. The engineering strains are plotted versus angle around the tube, where the weld seam was at approximately 0◦ . From this plot, it is seen that the maximum tensile bending strains in the corners are about 19%, which are lower than the strains seen in the 6 mm highexpansion die (approximately 30%). Fig. 21 shows the predicted and measured results for the HSLA350-T1.8 mm tube hydroformed using the highexpansion process with a corner radius of 6 mm. From this plot, it is seen that the maximum tensile bending strains in the corners are approximately 35%. Tube thickness measurements were performed on the hydroformed tubes using a non-destructive, ultrasonic measurement device. These measurements were then compared to values obtained from numerical simulations. All thickness measurements were taken at the half-length of the tube. Fig. 22 shows the measured (experimental) and predicted (numerical) tube thickness results for the DP600-T1.8 mm tube formed using the high-expansion process with a corner radius of 6 mm. The results are plotted versus angle around the tube, where the weld seam was at approximately 0◦ . Fig. 23 shows the tube wall thickness measurements for the DP780T1.5 mm material formed using the high-expansion process with a corner radius of 18 mm. Fig. 24 shows the tube wall thickness measurements for the HSLA350-T1.8 mm material formed using the high-expansion process with a corner radius of 6 mm.
Fig. 26 – Contour plot of percent thickness change for the DP780-T1.5 mm tube formed using the high-expansion hydroforming process with the 18 mm corner radius die. Maximum thickness reduction at 10.3%.
Fig. 27 – Contour plot of percent thickness change for the HSLA350-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die. Maximum thickness reduction at 11.7%. Figure shows tube after a trimming process at a length of 400 mm.
From the findings comparing numerical and experimental results for multiple tubes, it is evident that the numerical model simulations used in this research are capable of accurately predicting the material behaviour during the hydroforming process.
5.2.
Fig. 25 – Contour plot of percent thickness change for the DP600-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die. Maximum thickness reduction at 10.0%. Figure shows tube after a trimming process at a length of 400 mm.
Numerical predictions
Fig. 25 shows a sample result of the numerical simulation of the high-expansion hydroforming process for the DP600T1.8 mm material formed with the 6 mm corner radius die using the optimized forming profiles. The figure shows the predicted contours of the percent change in thickness where the largest thickness reduction was about 10.0%. Table 2 lists the predicted maximum percentage of thickness reduction for all materials that were hydroformed using the high-expansion process with the 6 mm corner radius die. Fig. 26 shows the contours of the percent reduction in thickness of the high-expansion hydroforming process for the DP780-T1.5 mm material formed with the 18 mm corner radius
122
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 28 – Predicted tube thickness distribution around the perimeter of two sample tubes hydroformed using the low- and high-expansion process with the 6 and 18 mm dies. Numerical measurements taken at half-length of the tube.
die. The figure shows that the largest thickness reduction was about 10.2%. Fig. 27 shows the contours of the percent reduction in thickness of the high-expansion hydroforming process for the HSLA350-T1.5 mm material formed with the 6 mm corner radius die. The figure shows that the largest thickness reduction was approximately 11.7%. Table 3 lists the predicted maximum percentage of thickness reduction for all materials that were formed using the high-expansion process with the 18 mm corner radius die. Fig. 28 shows the predicted tube thickness distribution around the perimeter of four sample tubes hydroformed using the low- and the high-expansion process with both the 6 and 18 mm corner radius dies. The results shown are for the DP600T1.8 mm material. The predicted thickness data were taken at the half-length of the tube and were plotted versus angle around the tube, where the weld seam was at approximately 0◦ . As seen from the graph, the high-expansion hydroforming process caused a high level of tube thinning when compared to the low-expansion process. In the high-expansion case, the maximum percentage change in tube thickness due to hydroforming (from a nominal tube thickness of 1.8 mm) was 10.0% for the 6 mm corner radius case and 11.5% for the 18 mm corner radius case (compared to only 2.8–3.3% for the low-expansion cases). Experimental verification of the tube
Table 3 – Predicted maximum percentage of thickness change for all materials hydroformed using the high-expansion die with a corner radius of 18 mm Material DDQ-T1.8 mm HSLA350-T1.5 mm HSLA350-T1.8 mm DP600-T1.8 mm DP780-T1.5 mm
Maximum (%) thickness reduction 10.0 12.0 12.4 11.5 10.3
thickness results was carried out using the ultrasonic measurement device which measured the thickness changes after hydroforming. The measured results for the high-pressure process closely matched the predicted results (see Figs. 22–24). Comparing the results for the high-expansion process for the two corner radii as shown in Tables 2 and 3, it is evident that there is little difference in the percentage of thickness change between the two corner radii. The amount of thickness change in the high-pressure process for both insert sizes, i.e. 6 and 18 mm, depends on the amount of end-feeding used to hydroform the tubes (shown in Fig. 14 for the 6 mm insert and Fig. 15 for the 18 mm insert). Therefore, even though the 6 mm insert would ordinarily cause higher expansion of the tube, and, consequently, a higher tube thickness change would be expected in the 6 mm die as compared to the 18 mm insert, the amount of end-feeding used to hydroform the tube in the 6 mm insert is higher than the 18 mm insert, which tends to compensate for the loss of thickness.
6.
Conclusions
The tube hydroforming process offers many advantages, especially for the automotive industry, in achieving part reduction and more efficient use of material. Advanced high-strength steels and dual phase steels are attractive candidate materials, offering higher strengths for energy absorption and weight reduction. Tube hydroforming of these high-strength steels, however, especially in the high-expansion processes, is difficult. Trial-and-error methods generally used to hydroform tubes are both expensive and time-consuming, and there is no guarantee that optimum forming profiles will be found. In this research, an optimization method linked with the finite element analysis which employed forming limit diagrams as a failure prediction tool has been successfully used to develop both the fluid pressure and end-feed profiles required to hydroform tubes made from different materials. The forming profiles developed through the optimization procedure were used to successfully hydroform multiple tubes
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
for all materials in this research without any rupture or wrinkling occurring.
Acknowledgments Financial support for this research from General Motors of Canada Limited, Dofasco, the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Ontario Research and Development Challenge Fund is gratefully acknowledged. Also the authors would like to thank Red Cedar Technology for the use of the optimization program HEEDS.
references
Abedrabbo, N., Pourboghrat, F., Carsley, J., 2006. Forming of aluminum alloys at elevated temperatures. Part 2. Numerical modeling and experimental verification. Int. J. Plast. 22 (2), 342–737. Abedrabbo, N., Zafar, N., Averill, R., Pourboghrat, F., Sidhu, R., 2004. Optimization of a tube hydroforming process. In: S. Ghosh (Ed.), NUMIFORM2004, AIP Conf. Proc., vol. 712, OH, USA. Dohmann, F., Hartl, C., 1996. Hydroforming–a method to manufacture lightweight parts. J. Mater. Process. Technol. 60, 669–676. Fann, K.-J., Hsiao, P.-Y., 2003. Optimization of loading conditions for tube hydroforming. J. Mater. Process. Technol. 140, 520–524. Ferrier, J., 1996. Hydroforming paradigms. In: Innovations in Hydroforming Technology. Huber and Bauer Inc. Gao, L., Motsch, S., Strano, M., 2002. Classification and analysis of tube hydroforming processes with respect to adaptive FEM simulations. J. Mater. Process. Technol. 129, 261–267. Gorni, A.A., Mei, P.R., 2005. Development of alternative as-rolled alloys to replace quenched and tempered steels with tensile
123
strength in the range of 600–800 MPa. J. Mater. Process. Technol. 162–163, 298–303. Hallquist, J.O., 2006. LS-DYNA Theoretical Manual. Livermore Software Technology Corporation, http://www.lstc.com/. HEEDS (Hierarchical Evolutionary Engineering Design System), 2008. Getting Started Manual. Red Cedar Technology, MI, USA, http://www.redcedartech.com/. Imaninejad, M., Subhash, G., Loukus, A., 2005. Loading path optimization of tube hydroforming process. Int. J. Mach. Tools Manuf. 45 (12–13), 1504–1514. Jansson, M., Nilsson, L., Simonsson, K., 2007. On process parameter estimation for the tube hydroforming process. J. Mater. Process. Technol. 190 (1–3), 1–11. Johnson, K., Nguyen, B., Davies, R., Grant, G., Khaleel, M., 2004. A numerical process control method for circular tube hydroforming prediction. Int. J. Plast. 20, 1111–1137. 2007. LS-DYNA Keyword Users Manual, Version 971. LSTC. Manabe, K., Suetake, M., Koyama, H., Yang, M., 2006. Hydroforming process optimization of aluminum alloy tube using intelligent control technique. Int. J. Mach. Tools Manuf. 46 (11), 1207–1211. Ni, C.-M., 1994. Stamping and hydroforming process simulation with a 3D finite element code. SAE Technical Paper 940753. Simo, J.C., Ortiz, M., 1985. A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations. Comput. Methods Appl. Mech. Eng. 49, 221–245. Stoughton, T.B., 2001. Stress-based forming limits in sheet metal forming. J. Eng. Mater. Technol. 123, 417–422. Stoughton, T., Zhu, X., 2004. Review of theoretical models of the strain-based FLD and their relevance to the stress-based FLD. Int. J. Plast. 20 (8–9), 1463–1486. Wu, L., Yu, Y., 1996. Computer simulations of forming automotive structural parts by hydroforming process. In: Proceedings of Numisheet’96. Yingyot, A.-U.-L., Gracious, N., Taylan Altan, 2004. Optimizing tube hydroforming using process simulation and experimental verification. J. Mater. Process. Technol. 146, 137–143.
journal homepage: www.elsevier.com/locate/jmatprotec
Optimization methods for the tube hydroforming process applied to advanced high-strength steels with experimental verification Nader Abedrabbo a,∗ , Michael Worswick a , Robert Mayer b , Isadora van Riemsdijk c a
Department of Mechanical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada b General Motors Research & Development Center, Warren, MI 48090, United States c Dofasco Inc. (ArcelorMittal) Research & Development, Hamilton, ON L8N 3J5, Canada
a r t i c l e
i n f o
a b s t r a c t
Article history:
In this paper, an optimization method linked with the finite element method is presented for
Received 20 September 2007
developing forming parameters of the tube hydroforming (THF) process for several advanced
Received in revised form
high-strength steel (AHSS) materials. The goal of this research was to maximize formability
17 January 2008
by identifying the optimal internal hydraulic pressure and end-feed rate, while satisfying
Accepted 19 January 2008
the failure limits defined by the forming limit diagram (FLD). The optimization software HEEDS was used in combination with the nonlinear structural finite element code LS-DYNA to carry out the investigation. The pressure and feed profiles identified through the auto-
Keywords:
mated optimization procedures were obtained with accuracy and with no trial-and-error
Optimization
experiments. Experimental validation of the optimized profiles by hydroforming multiple
Tube hydroforming
tubes made from different materials was then performed successfully. © 2008 Elsevier B.V. All rights reserved.
HEEDS LS-DYNA Forming limit diagrams High-strength steels Formability
1.
Introduction
Weight reduction is a key priority for improving automotive fuel economy. Multiple candidates for replacing mild steel in automotive structures have been proposed, e.g. advanced high-strength steels (AHSS), aluminum, magnesium, and composites. Advanced high-strength steels, in particular, are attractive candidate materials, offering higher strength for energy absorption and the opportunity to reduce weight through use of thinner gauges. However, though steels in general are highly formable, the increase in strength achieved by
∗
the AHSS materials is associated with a partial compromise in formability. In recent years, tube hydroforming (THF) technology has become a popular method, especially in the automotive industry, for producing complex three-dimensional structural shapes because of its enormous advantages over the more traditional processes. There are several advantages to hydroforming over conventional processes such as stamping and welding. These advantages include part consolidation, weight reduction due to improved part design, improved structural strength and stiffness, and reduction in the associated
Corresponding author. Tel.: +1 519 885 1211x35887; fax: +1 519 885 5862. E-mail address: [email protected] (N. Abedrabbo). 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.01.060
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
tooling and material costs (Dohmann and Hartl, 1996). In view of these advantages, the range of parts currently being produced or developed using THF by the automotive industry continues to grow. These parts include engine cradles, radiator supports, roof side rails, exhaust manifold assemblies, camshafts, sub-frames and instrument support panels (Ferrier, 1996; Ni, 1994). Two methods are widely used to hydroform circular tubes: low-expansion (low-pressure) and high-expansion (high-pressure) hydroforming processes. A key difference between the two processes is that the low-expansion process introduces only limited circumferential expansion with no end-feeding required to form the tube. In the high-expansion method, the circumferential strains are large, causing excessive tube thinning to occur and therefore end-feeding of the tube, i.e. extra tube material pushed into the die, is used to counter the thinning problem. Tubes formed using the lowexpansion method are generally easy to manufacture, with very little failure or wrinkling problems occurring (or none at all). The success of the high-expansion process, on the other hand, is highly dependent on a number of variables, including material formability, friction conditions, and most importantly, the loading paths (pressure and end-feeding) used to form the tube. Gao et al. (2002) classified the tube hydroforming process based on loading parameters and concluded that the pressure dominant process to be the most challenging load curve determination problem since it involves a high risk for wrinkling, bursting and leakage. Therefore, parameters which are derived from this process would be the most beneficial for load curve optimization (Jansson et al., 2007). The loading paths in the THF process are traditionally determined using trial-and-error procedures and, in particular, rely on past experiences of the operator (Fann and Hsiao, 2003). The THF process is further complicated if new materials and die geometries are used for which the operator has no prior knowledge. The traditional process is both timeconsuming and expensive. Furthermore, there is no guarantee that “optimum” pressure and end-feed profiles can be found by the trial-and-error process. Therefore, in order to determine the pressure and end-feed hydroforming profiles for any material and die geometry, an integrated approach to the problem comprised of the finite element analysis (FEA) of the hydroforming process, a failure model, and an optimization code is required. Numerical analysis is a critical tool for understanding the complex deformation mechanics that occur during sheet forming processes. Finite element analysis and simulations are used in automotive design and formability processes to predict deformation behaviour accurately during stamping operations. Finite element method (FEM) simulation of the hydroforming process has been proven to be an advantageous tool in assisting automotive designs. Ni (1994) and Wu and Yu (1996) simulated engine cradle components. Over the years, several research papers have been published about the optimization of the tube hydroforming process. In these papers, multiple optimization techniques have been proposed. Johnson et al. (2004) proposed a method in which the optimization algorithm uses the material stress–strain curve and the deformation theory of plasticity to relate the current stress and strain increments to the
111
next applied load increments. In this method, a controlled increment in plastic strain was prescribed for the next solution increment, and the pressure and end-feed increments were calculated to give a constant ratio of incremental axial and hoop strains. Imaninejad et al. (2005) used LS-DYNA and the optimization software LS-OPT to optimize the internal pressure and axial feed of a T-joint design in which a minimum thickness variation was chosen as a design objective and maintaining the effective stress below the materials ultimate stress. Fann and Hsiao (2003) used the conjugate gradient optimization methods to optimize a T-shaped tube using LS-DYNA. Yingyot et al. (2004) investigated two methods; self-feeding and adaptive simulation techniques, in order to optimize the tube hydroforming process and experimental versus numerical results were presented. Manabe et al. (2006) studied the optimization process of tube hydroforming using a virtual-database-assisted fuzzy control system to determine the forming parameters. Also, Jansson et al. (2007) proposed an adaptive optimization method based on the use of the response surface methods. In the current research, the optimization software Hierarchical Evolutionary Engineering Design System (HEEDS), which uses genetic algorithm (GA) search methods and utilizes a unique search strategy called Simultaneous Hybrid Exploration that is Robust, Progressive and Adaptive (SHERPA), which is an innovative method that uses multiple search methods simultaneously (not sequentially), was used in combination with the nonlinear structural finite element code LS-DYNA (Hallquist, 2006) to optimize the process parameters and determine the best loading paths for the tube hydroforming process.
2.
THF experiments
2.1.
Experimental setup
Tube hydroforming of circular tubes was performed using square-shaped dies with two different corner radii inserts: 6 and 18 mm. Tubes were hydroformed using a 1000-tonnes (9806 kN) macrodyne hydroforming press at the University of Waterloo Labs, shown in Fig. 1. Fig. 2 shows a sectional view of the high-expansion die used to hydroform the tubes with the 18 mm corner radius insert attached. Fig. 3 shows a cross-section schematic of the 6 mm die insert used in the research with die dimensions. The sloped area shown in Fig. 3 provides a smooth transition between the round tube section and the final square die shape. To illustrate the differences between the two types of hydroforming processes usually used in THF, Fig. 4 shows a comparison between the low- and high-expansion tubes at the end of the hydroforming process. Also shown in the graph is the original tube size used in the two types of experiments before hydroforming. In the low-expansion hydroforming process, the initial tube size is larger than the die. As the die closes, it mechanically forms the tube to the die shape with a small fluid pressure present to prevent buckling. After the die is closed, the fluid pressure is gradually increased to form the tube. In the high-expansion hydroforming process, the tube is initially in contact with the sides of the die, as shown. As fluid
112
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 3 – Cross-section of the 6 mm insert showing dimensions (mm).
Fig. 1 – The macrodyne 1000 tonnes (9806 kN) forming press at the University of Waterloo.
In order for the tubes to be hydroformed using the highexpansion process, i.e. to counter tube thinning due to the higher expansion, end-feeding was introduced using two endfeed actuators, each with 1112 kN (250 kip) force at each end of the tube. End-feeding pushes material into the die, thereby increasing the formability of the tubes. One of the end-feed actuators used in the high-expansion experiments is shown in Fig. 5. In order to seal the tube during hydroforming, the tip of the end-feed actuators, which incorporates a Teflon Oring and a high-pressure polymeric seal, is inserted inside the tube a distance of 70 mm where the edge of the tube sits on a shoulder. The tip of the actuator used in the experiments showing the sealing O-rings are shown in Fig. 6.
2.2.
Experimental procedure
Tubes were cut to a length of 785 mm and the ends of the tubes were deburred to prevent damage to the Teflon and polymeric O-rings. All tube specimens were lubricated using a solid-film lubricant, HydroDraw 625, which was first sprayed onto the tube and allowed to dry. Twist-compression tests with the die steel (P20) and sheet steels of the tube materials were performed. The Coulomb’s coefficient of friction was determined
Fig. 2 – High-expansion hydroforming top and bottom dies with the 18 mm insert attached.
pressure is increased, tube expansion occurs to fill the corner region. In this case, no mechanical forming occurs before fluid pressure is applied. As seen in Fig. 4, the high-expansion process causes higher expansion of the tube (between 18 and 26% change in perimeter depending on corner radius) compared to the low-expansion process (approximately 0.8% change in perimeter).
Fig. 4 – Comparison between the original round tube size (NH) and the low- and high-expansion hydroforming processes for the 6 mm corner radius dies. The small square is the final low-expansion tube (LP), while the larger square is for the high-expansion (HP) process tube.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 5 – One of the 1112 kN (250 kip) end-feed actuators used in the high-expansion hydroforming experiments.
Fig. 6 – Tip of the end-feed actuator showing the Teflon ring and polymeric O-ring used to seal the tube. The tube rests on the shoulder during the hydroforming process.
to be approximately 0.05, which was used in the numerical analysis. After placing the tubes in the die cavity and closing the dies, the end-feed actuators were advanced to allow the tube ends to rest on the shoulders of the end-feeds. Air was removed using a vacuum pump, and the tube was then filled with water. Next, the loading paths, i.e. end-feed displacement versus time and fluid pressure versus time, were applied to hydroform the tube. The LabView program was used to control the process variables. In order to ensure accurate pressure application for every end-feed displacement point, a pressure versus displacement profile was used to control the process. If the system response time was slow, or in case of intermittent pressure drop, the system stopped the end-feed actuators from advancing until the pressure reaches the required level, after which the hydroforming process continues. This interactive procedure prevented the tube from either wrinkling (due to excessive feed) or tearing (due to high-pressure).
2.3.
113
Fig. 7 – A sample of a tube formed using the high-expansion process with non-optimized forming profiles using the 6 mm die. Excessive thinning caused the tube to burst before it formed to the die shape.
tube to untimely burst. In this case, the amount of extra tube material pushed into the die to counter the thinning effects was not enough to prevent tube bursting. Fig. 7 shows a sample of a tube hydroformed using the high-expansion process with a minimum amount of end-feeding for which tube burst occurred. As the pressure in the tube increased, the tube began to expand from the middle of the tube, starting at the regions of the tube in contact with die (see Fig. 4) and continuing into the unsupported regions at the corners, causing the material to thin. Expansion continued until the excessive thinning in the material caused the tube to burst before the tube formed to the die’s final shape. In the second case, to overcome the excessive thinning problem, a certain amount of end-feeding of the tube was used. However, the amount of end-feeding required for each tube depended on the geometry of the die (6 or 18 mm corner radius), friction conditions, and the type of material being hydroformed. Therefore, end-feeding of each tube material was determined such that enough material was fed into the die to prevent excessive thinning. It is important to note, however, that material overfeeding will cause the longitudinal compressive stress to be higher than the hoop tensile stress in the material, causing undesirable buckling or wrinkling in the tube. Fig. 8 depicts the case of overfeeding. Although the tube was formed without bursting, the extra material pushed into the die caused buckling in the tube along the corner radius where the material is unsupported and free to buckle.
THF using non-optimized forming profiles
To illustrate the importance of optimizing the forming profiles, two cases are shown next where non-optimized hydroforming profiles were used to form the tubes. In the first case, the tubes were formed with a minimal amount of end-feeding. Due to a combination of expansion and friction between the die and the tube surfaces, tube thinning occurred, causing the
Fig. 8 – A tube formed using the high-expansion hydroforming process with excessive amount of end-feeding. The extra material pushed in caused the tube to buckle as shown by the area in the dashed line.
114
3.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Materials
In the current research, monolithic tubes made from multiple materials and wall thicknesses were considered. Materials used include the following: conventional deep drawing quality (DDQ) steel of wall thickness 1.8 mm; high-strength low alloy (HSLA-350) steel with wall thicknesses of 1.5 and 1.8 mm; and AHSS materials comprising the dual phase (DP) alloys DP600 with wall thickness of 1.8 mm and DP780 with wall thickness of 1.5 mm. High-strength low alloy and DP steels are currently being used to reduce the weight of automobiles (Gorni and Mei, 2005). Not only do they display good formability and weldability, but also they benefit from the high-strength of martensite with reduced alloying and without the costly inclusion of postfabrication heat treatment. All tubes in this research had an outer diameter of 76.2 mm.
3.1.
Constitutive equations (flow stress)
Finite element analysis and simulations are vital tools in understanding the deformation behaviour that occurs during forming and crash experimental operations. Confidence in the numerical analysis depends on the use of a constitutive model that accurately describes the behaviour of the material being investigated. Therefore, in this research, full material characterization was performed on the different materials used in order to extract the required parameters of the constitutive models to be used in the numerical analysis. The materials were tested in both the sheet and the as-formed tube conditions in order to identify tube forming operations’ effects on tube properties. For the as-formed tube material, since rolling operations used to manufacture the tubes induce varying levels of workhardening around the circumference of the tube, testing was performed on specimens from three positions on the perimeter of the tube, corresponding to the 3, 6, and 9 O’clock positions (the weld seam is located at 12 O’clock) as shown in Fig. 9. From these experimental tensile tests, the parameters of the Power-Law constitutive model (flow stress) were fit to the experimental results for each material. The Power-Law model
is represented as follows: (¯ ¯ εp ) = K(¯εp + ε0 )
n
(1)
where “K” (strength-hardening coefficient) and “n” (strainhardening exponent) are material constants. ε¯ p is the effective plastic strain and ε0 is a constant representing the elastic strain to yield. Although multiple tensile tests were performed for each material in different directions (sheet data, 3, 6, and 9 O’clock tube positions), and the constitutive parameters were calculated for each direction, only one set of parameters is used in the numerical analysis. In order to represent the tube in its multiple directions in the numerical analysis, a weighted average method was used to calculate the Power-Law constitutive parameters. The following formula was used: X avg =
2(X3 O clock + X6 O clock ) 3
(2)
where X represents a general variable. This specific averaging method was used because tensile test results indicated that the work-hardening that is imparted to the tube during the tube manufacturing process is symmetric but not uniform. Therefore, the 3 and 9 O’clock positions were found to be very close and thus considered equal. Table 1 lists the constitutive parameters of the Power-Law model for all the materials used in this research.
3.2.
Failure criteria
In order to determine the required hydroforming profiles using the combination of the optimization program and the FEA analysis, a failure criteria for the different materials must be determined. Including the failure criteria in the FEA program helps to identify both the time and the location of failure in the numerical model. In the current project, failure prediction was based on the stress-based forming limit diagrams (-FLD). In forming processes, if the loading path is sufficiently close to linear (proportional loading) then a general strain-based FLD can be used to assess failure of the part. However, for a general forming process in which the loading path may not be linear (non-proportional loading), it becomes necessary to determine the strain-based FLD for each element based on its loading path. This process is computationally expensive, and therefore, a path-independent failure method should be used instead. Stoughton (2001) proposed a method to transform the path-dependent strain-based FLD into a pathindependent stress-based FLD. A review of different types of
Table 1 – Constitutive parameters for the different tubular materials used
Fig. 9 – Top view of cross-section of tube showing locations of test samples used in tensile test and also showing the weld seam location.
Material
t0 (mm)
OD (mm)
K (MPa)
n
ε0
DDQ HSLA-350 HSLA-350 DP-600 DP-780
1.8 1.5 1.8 1.8 1.5
76.2 76.2 76.2 76.2 76.2
578.14 684.00 679.88 900.00 1166.37
0.183 0.095 0.121 0.109 0.130
7.46E-4 1.81E-3 1.49E-3 2.24E-3 2.60E-3
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
115
and satisfying the stress-based forming limit diagram (-FLD). The optimization problem can be defined as follows: Maximize:
Displacement of node “n” (Fig. 11)
Subject to:
All elements of tube model to lie below -FLD
By varying:
Fig. 10 – Stress-based forming limit diagrams (-FLDs) for all materials.
-FLD and their use in FEA can be found in Stoughton and Zhu (2004). Fig. 10 shows the -FLD for all the materials used in this research which were determined from the -FLD’s as described by Stoughton (2001).
4.
THF process optimization
4.1.
Problem definition
Pressure:
Pmin < Pi < Pmax
End-feed force:
Fmin < Fi < Fmax , where i:1–10 (depending on curve profile segments)
Pmax and Fmax represent the maximum pressure and end-feed force possible in the current hydroforming press, respectively. The profiles were setup to increase in a monotonic fashion. In the current setup, since the end-feed actuators used in the experiments have a maximum force of 1112 kN (250 kip), the best method to ensure that the results obtained from the numerical simulations satisfy this limitation was to use the end-feed force as an optimization variable instead of the end-feed displacement. After the optimum solutions were found, the end-feed displacements to be used experimentally were then easily determined from the numerical end-feed displacement. The acquired profiles were then used in the experimental setup to successfully form all tubes (a minimum of six tubes were formed for each material).
4.2.
The optimization objective in the current research was to determine the corresponding internal fluid pressure schedule and the end-feed displacement that would allow the tube to expand to maximum extent without bursting or buckling. By considering a point “n” on the corner of the tube, shown in Fig. 11, as the tube expands into the die, the distance “U” traveled by the point “n” increases. The optimal rates and values of axial feed and internal pressure are determined so as to maximize the distance ‘U’ while avoiding severe thinning
Fig. 11 – Schematic of the optimization process. “n” is a center node, “P” is hydroforming pressure, and “U” is node displacement.
HEEDS optimization program
The optimization software HEEDS was used in combination with the nonlinear structural finite element code LS-DYNA to carry out the numerical optimization. The characteristics of the design space associated with the current optimization problem were not known a priori. In this case, it was advisable to employ a combination of global and local search techniques in order to achieve a broad and effective search for an optimal solution. For such problems, HEEDS utilizes a combination of evolutionary, gradient-based, and design-ofexperiments search heuristics (HEEDS, 2008). By intelligently coupling global and local search techniques, the HEEDS optimization algorithms are able to find excellent solutions to even the most challenging design problems. Local optimization methods (e.g., nonlinear sequential programming, response surface methods, etc.) are valuable for fine-tuning a design, but not for exploring different design concepts in an effort to identify a much better design. Because the mathematical cost or objective functions associated with many practical design problems are multi-modal (i.e., they have many peaks and valleys) or even discontinuous, the use of global search methods (e.g., parallel genetic algorithms) improves the likelihood of achieving significant design innovation. While global methods search broadly over a large design space, local optimization methods simultaneously focus on promising sub-regions of the design space to identify the best designs in that region. HEEDS applies several optimization methods simultaneously, allowing each method to take advantage of the best attributes and solutions found from other parallel searches. The multiple semi-independent search processes exchange
116
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
information about the solution space with each other, helping to jointly satisfy multiple constraints and objectives. This search method is called a heterogeneous multi-agent approach. This approach quickly identifies design attributes with good potential and uses them to focus, improve and accelerate the search for an optimum solution. More details about the HEEDS program can be found at www.redcedartech.com.
4.2.1.
Genetic algorithm
HEEDS employs a GA to perform evolutionary search. GAs are particularly useful when the design space is large and complex. The main problem with using a simple GA is the potentially large number of design evaluations required to obtain a set of satisfactory solutions. HEEDS reduces the number of evaluations required to obtain a set of satisfactory solutions by hierarchically decomposing a problem with multiple agents that represent the problem in various ways, while combining efficient local search methods (e.g., response surface methods, nonlinear sequential quadratic programming, and simulated annealing). Genetic algorithm is a search procedure that is based on the mechanics of natural selection. Specific knowledge is embedded in a chromosome (or design vector) which represents a possible design with a set of values of all the design variables. The number of choices per design variable determines the fidelity (or resolution) of each design variable. These design variables are the building blocks used to construct a particular design. The GA creates and destroys designs during a process that involves decoding each chromosome, evaluating its satisfaction of constraints and its performance relative to the objectives then allowing a simulated “natural selection” to determine which designs are eliminated and which survive to generate other derivative designs. Designs that perform well (relative to constraints and objectives) have a higher probability of surviving to influence future designs (their “offspring”). During reproduction, the two genetic operators commonly modeled that produce new chromosomes (or design vectors) are called crossover and mutation.
4.2.2.
Crossover
The crossover operation (sometimes also called “recombination”) forms a new solution by combining parts of two existing solutions. A high crossover rate (fraction of population replaced by crossover during one generation of reproduction) will produce many new designs in each generation, but will also have a high probability of disrupting (and potentially losing, at least temporarily) higher-performance designs already found, and requires more evaluations of constraints and objectives in each generation which are typically the most costly computing operation in the entire problem.
4.2.3.
resemble each other, thereby making it difficult for crossover to generate solutions that differ very much from the current set). A set of co-existing designs defines a population, while successive populations are termed generations. That is, each period during which a set of existing solutions are evaluated then used with natural selection, crossover, and mutation to generate a new set of solutions, is called a generation. A large population typically contains more genetic diversity (i.e., different values of design variables) that typically improves the ultimate results of the GA search. However, the more new individuals created in each generation, the more computer time must be spent evaluating the constraints and objectives of the new individuals, so there is a tradeoff that must be made. Within each agent, a GA search begins by creating a single initial population, wherein chromosomes (vectors of design variable values) are randomly created. At this point the performance (constraint satisfaction) and objective values of each design is evaluated. Biased by the evaluations obtained, the GA uses unary (mutation) and binary (crossover) operators on these designs to create another population. This population probabilistically maintains the previously high performing designs while discarding poorly performing designs. New population members are evaluated and then additional rounds of generation and selection are performed. This process is repeated until satisfactory solution(s) are obtained. Incorporating these processes in a computer routine produces an algorithm that solves problems in a manner reminiscent of natural evolution. Independent GA searches in several agents can share information with each other through a user-defined migration process.
4.3.
FE numerical model
The explicit part of the finite element program LS-DYNA was used in the current research. To take advantage of process symmetry, one-eighth of the tube is considered in the FEM optimization model to reduce the simulation time, as shown in Fig. 12. The die and end-feeds were created using rigid shell elements of 2.0 mm × 2.0 mm size with material properties of steel. The tube material was created using 2.0 mm × 2.0 mm Belytschso-Tsay (LS-DYNA Manual, 2007) shell elements.
Mutation
Mutation is a reproduction operation that produces a new solution from a single existing solution through any of several ways. Mutation can change the value of one design variable or of many simultaneously and can change them in uniform random ways, or by a normal distribution (for example around the current values of the design variables). Mutation helps to maintain diversity and reduces the possibility of premature convergence (the tendency of a set of solutions to come closely
Fig. 12 – One-eighth FEM model of the tube hydroforming process used in the optimization process.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 13 – Process flow schematic of the optimization process showing the interaction between HEEDS and LS-DYNA.
117
Fig. 16 – Experimental results for DP600-T1.8 mm tube formed with the high-expansion process and the 6 mm corner radius die using the optimized forming profiles in Fig. 14. Figure shows tube after a trimming process at a length of 400 mm.
Fig. 17 – Experimental results for DP780-T1.5 mm tube formed with the high-expansion process and the 18 mm corner radius die using the optimized forming profiles in Fig. 15. Figure shows tube after a trimming process at a length of 400 mm.
Fig. 14 – Optimized hydroforming profiles for all materials hydroformed using the 6 mm high-expansion die.
Fig. 18 – Experimental results for HSLA350-T1.8 mm tube formed with the high-expansion process and the 6 mm corner radius die using the optimized forming profiles in Fig. 14. Figure shows tube after a trimming process at a length of 400 mm.
Fig. 15 – Optimized hydroforming profiles for all materials hydroformed using the 18 mm high-expansion die.
The user material subroutine (UMAT) option in LS-DYNA was used in this research where the von-Mises constitutive material model was implemented using the cutting-plane algorithm (Simo and Ortiz, 1985; Abedrabbo et al., 2006), which is based on the incremental theory of plasticity. The UMAT procedure was used in order to include the failure criteria into the FEM analysis. The stress-based forming limit curves (-FLD), as shown in Fig. 10, were represented in mathematical form using curve-fitting methods. During the simulation process, each element of the tube was compared
118
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 19 – Predicted and measured engineering strains for DP600-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die.
against the -FLD at every integration point to check whether it failed or not. If failure of an element occurred, an output file was written that contained the displacement of a node “n” on the corner of the tube (see Fig. 11). This output file was then used by HEEDS in the calculation of the next optimized step. The process continued until a suitable solution was reached and the tube was completely formed, i.e. node “n” contacted die at corner, and no element in the tube had failed. A fullscale model of the tube and dies was also implemented in order to compare the numerical results to the experimental ones.
4.4.
HEEDS–LS-DYNA interface
HEEDS is a fully automated optimization program that is modular, generic, and independent. The program can be linked to
any type of solver (commercial or otherwise). HEEDS uses the input file(s) and output file(s) from a baseline (initial) design to identify and analyze design candidates and generate project responses. A basic analysis definition in HEEDS consists of (Abedrabbo et al., 2004):
• A command that executes the solver to be used for the analysis of the design (LS-DYNA in this case). • All input files containing the optimization process variables that are varied for each evaluation (fluid pressure and endfeed force in this project). • At least one output file containing the responses of interest from a successful analysis run which are used to evaluate the performance of the design (node displacement in this case).
Fig. 20 – Predicted and measured strains for DP780-T1.5 mm tube formed using the high-expansion hydroforming process with the 18 mm corner radius die.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
119
Fig. 21 – Predicted and measured engineering strains for HSLA350-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die.
The optimization process and the interaction between the optimization program HEEDS and the FEA code LS-DYNA are represented in the flow schematic of Fig. 13.
in the 6 mm die as compared to the 18 mm die, and therefore a higher level of end-feed is required to prevent the tube from bursting.
4.5.
5. Experimental verification and numerical simulations
Optimization results
Fig. 14 shows the pressure versus end-feed displacement profiles acquired from the optimization process for all the tubes hydroformed using the 6 mm corner radius high-expansion die. Fig. 15 shows the optimization results for all the tubes hydroformed using the 18 mm corner radius high-expansion die. As seen from these two figures, the 6 mm corner radius die requires a higher amount of end-feeding to form the tubes without bursting as compared to the 18 mm corner radius die. This is anticipated since a higher level of tube thinning occurs
5.1.
Experimental validation
The fluid pressure and end-feed profiles determined for each material using the optimization methods described earlier were successfully used to hydroform tubes for all the different materials (as shown in Table 1) using the dies with corner radii of 6 and 18 mm. In total, there were 10 high-expansion numerical simulations performed; 5 for the 6 mm corner radius die
Fig. 22 – Measured and predicted tube wall thickness for DP600-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die.
120
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 23 – Measured and predicted tube wall thickness for DP780-T1.5 mm tube formed using the high-expansion hydroforming process with the 18 mm corner radius die.
and 5 for the 18 mm corner radius die. Experimentally, at least 6 tubes were hydroformed for each corresponding numerical simulation, for a total of 60 tubes. Fig. 16 shows a sample experimental result, after a trimming process, for the DP600-T1.8 mm tube formed with the high-expansion process and the 6 mm corner radius die using the optimized forming profile shown in Fig. 14 for the same material. Fig. 17 shows a sample experimental result, after a trimming process, for the DP780-T1.5 mm tube formed with the high-expansion process and the 18 mm corner radius die using the optimized forming profile shown in Fig. 15 for the DP780 material. Fig. 18 shows another sample experimental result, after a trimming process, for the HSLA350-T1.8 mm tube formed with the high-expansion process and the 6 mm corner radius die using the optimized forming profile shown in Fig. 14 for the same material. All the tubes in this research were experimentally hydroformed without any defects and
with no trial-and-error similar to the three samples shown in Figs. 16–18. The engineering strains within the formed tubes were measured by comparing changes in the shape of circular grids that were etched onto the tube prior to the tube hydroforming operation. These measurements were then compared to values obtained from numerical simulations. All strain measurements were taken at the half-length of the tube. Fig. 19 shows the predicted (numerical) and measured (experimental) engineering strains for the DP600-T1.8 mm tube formed using the high-expansion process with a corner radius of 6 mm. The engineering strains are plotted versus angle around the tube, where the weld seam was at approximately 0◦ . Experimental measurements were made for every other circle grid around the perimeter of the tube. Fig. 20 shows the predicted and measured results for the DP780-T1.5 mm tube hydroformed using the high-expansion
Fig. 24 – Measured and predicted tube wall thickness for HSLA350-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
121
Table 2 – Predicted maximum percentage of thickness change for all materials hydroformed using the high-expansion die with a corner radius of 6 mm Material DDQ-T1.8 mm HSLA350-T1.5 mm HSLA350-T1.8 mm DP600-T1.8 mm DP780-T1.5 mm
Maximum (%) thickness reduction 10.0 11.4 11.7 10.0 9.7
process with a corner radius of 18 mm. The engineering strains are plotted versus angle around the tube, where the weld seam was at approximately 0◦ . From this plot, it is seen that the maximum tensile bending strains in the corners are about 19%, which are lower than the strains seen in the 6 mm highexpansion die (approximately 30%). Fig. 21 shows the predicted and measured results for the HSLA350-T1.8 mm tube hydroformed using the highexpansion process with a corner radius of 6 mm. From this plot, it is seen that the maximum tensile bending strains in the corners are approximately 35%. Tube thickness measurements were performed on the hydroformed tubes using a non-destructive, ultrasonic measurement device. These measurements were then compared to values obtained from numerical simulations. All thickness measurements were taken at the half-length of the tube. Fig. 22 shows the measured (experimental) and predicted (numerical) tube thickness results for the DP600-T1.8 mm tube formed using the high-expansion process with a corner radius of 6 mm. The results are plotted versus angle around the tube, where the weld seam was at approximately 0◦ . Fig. 23 shows the tube wall thickness measurements for the DP780T1.5 mm material formed using the high-expansion process with a corner radius of 18 mm. Fig. 24 shows the tube wall thickness measurements for the HSLA350-T1.8 mm material formed using the high-expansion process with a corner radius of 6 mm.
Fig. 26 – Contour plot of percent thickness change for the DP780-T1.5 mm tube formed using the high-expansion hydroforming process with the 18 mm corner radius die. Maximum thickness reduction at 10.3%.
Fig. 27 – Contour plot of percent thickness change for the HSLA350-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die. Maximum thickness reduction at 11.7%. Figure shows tube after a trimming process at a length of 400 mm.
From the findings comparing numerical and experimental results for multiple tubes, it is evident that the numerical model simulations used in this research are capable of accurately predicting the material behaviour during the hydroforming process.
5.2.
Fig. 25 – Contour plot of percent thickness change for the DP600-T1.8 mm tube formed using the high-expansion hydroforming process with the 6 mm corner radius die. Maximum thickness reduction at 10.0%. Figure shows tube after a trimming process at a length of 400 mm.
Numerical predictions
Fig. 25 shows a sample result of the numerical simulation of the high-expansion hydroforming process for the DP600T1.8 mm material formed with the 6 mm corner radius die using the optimized forming profiles. The figure shows the predicted contours of the percent change in thickness where the largest thickness reduction was about 10.0%. Table 2 lists the predicted maximum percentage of thickness reduction for all materials that were hydroformed using the high-expansion process with the 6 mm corner radius die. Fig. 26 shows the contours of the percent reduction in thickness of the high-expansion hydroforming process for the DP780-T1.5 mm material formed with the 18 mm corner radius
122
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
Fig. 28 – Predicted tube thickness distribution around the perimeter of two sample tubes hydroformed using the low- and high-expansion process with the 6 and 18 mm dies. Numerical measurements taken at half-length of the tube.
die. The figure shows that the largest thickness reduction was about 10.2%. Fig. 27 shows the contours of the percent reduction in thickness of the high-expansion hydroforming process for the HSLA350-T1.5 mm material formed with the 6 mm corner radius die. The figure shows that the largest thickness reduction was approximately 11.7%. Table 3 lists the predicted maximum percentage of thickness reduction for all materials that were formed using the high-expansion process with the 18 mm corner radius die. Fig. 28 shows the predicted tube thickness distribution around the perimeter of four sample tubes hydroformed using the low- and the high-expansion process with both the 6 and 18 mm corner radius dies. The results shown are for the DP600T1.8 mm material. The predicted thickness data were taken at the half-length of the tube and were plotted versus angle around the tube, where the weld seam was at approximately 0◦ . As seen from the graph, the high-expansion hydroforming process caused a high level of tube thinning when compared to the low-expansion process. In the high-expansion case, the maximum percentage change in tube thickness due to hydroforming (from a nominal tube thickness of 1.8 mm) was 10.0% for the 6 mm corner radius case and 11.5% for the 18 mm corner radius case (compared to only 2.8–3.3% for the low-expansion cases). Experimental verification of the tube
Table 3 – Predicted maximum percentage of thickness change for all materials hydroformed using the high-expansion die with a corner radius of 18 mm Material DDQ-T1.8 mm HSLA350-T1.5 mm HSLA350-T1.8 mm DP600-T1.8 mm DP780-T1.5 mm
Maximum (%) thickness reduction 10.0 12.0 12.4 11.5 10.3
thickness results was carried out using the ultrasonic measurement device which measured the thickness changes after hydroforming. The measured results for the high-pressure process closely matched the predicted results (see Figs. 22–24). Comparing the results for the high-expansion process for the two corner radii as shown in Tables 2 and 3, it is evident that there is little difference in the percentage of thickness change between the two corner radii. The amount of thickness change in the high-pressure process for both insert sizes, i.e. 6 and 18 mm, depends on the amount of end-feeding used to hydroform the tubes (shown in Fig. 14 for the 6 mm insert and Fig. 15 for the 18 mm insert). Therefore, even though the 6 mm insert would ordinarily cause higher expansion of the tube, and, consequently, a higher tube thickness change would be expected in the 6 mm die as compared to the 18 mm insert, the amount of end-feeding used to hydroform the tube in the 6 mm insert is higher than the 18 mm insert, which tends to compensate for the loss of thickness.
6.
Conclusions
The tube hydroforming process offers many advantages, especially for the automotive industry, in achieving part reduction and more efficient use of material. Advanced high-strength steels and dual phase steels are attractive candidate materials, offering higher strengths for energy absorption and weight reduction. Tube hydroforming of these high-strength steels, however, especially in the high-expansion processes, is difficult. Trial-and-error methods generally used to hydroform tubes are both expensive and time-consuming, and there is no guarantee that optimum forming profiles will be found. In this research, an optimization method linked with the finite element analysis which employed forming limit diagrams as a failure prediction tool has been successfully used to develop both the fluid pressure and end-feed profiles required to hydroform tubes made from different materials. The forming profiles developed through the optimization procedure were used to successfully hydroform multiple tubes
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 110–123
for all materials in this research without any rupture or wrinkling occurring.
Acknowledgments Financial support for this research from General Motors of Canada Limited, Dofasco, the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Ontario Research and Development Challenge Fund is gratefully acknowledged. Also the authors would like to thank Red Cedar Technology for the use of the optimization program HEEDS.
references
Abedrabbo, N., Pourboghrat, F., Carsley, J., 2006. Forming of aluminum alloys at elevated temperatures. Part 2. Numerical modeling and experimental verification. Int. J. Plast. 22 (2), 342–737. Abedrabbo, N., Zafar, N., Averill, R., Pourboghrat, F., Sidhu, R., 2004. Optimization of a tube hydroforming process. In: S. Ghosh (Ed.), NUMIFORM2004, AIP Conf. Proc., vol. 712, OH, USA. Dohmann, F., Hartl, C., 1996. Hydroforming–a method to manufacture lightweight parts. J. Mater. Process. Technol. 60, 669–676. Fann, K.-J., Hsiao, P.-Y., 2003. Optimization of loading conditions for tube hydroforming. J. Mater. Process. Technol. 140, 520–524. Ferrier, J., 1996. Hydroforming paradigms. In: Innovations in Hydroforming Technology. Huber and Bauer Inc. Gao, L., Motsch, S., Strano, M., 2002. Classification and analysis of tube hydroforming processes with respect to adaptive FEM simulations. J. Mater. Process. Technol. 129, 261–267. Gorni, A.A., Mei, P.R., 2005. Development of alternative as-rolled alloys to replace quenched and tempered steels with tensile
123
strength in the range of 600–800 MPa. J. Mater. Process. Technol. 162–163, 298–303. Hallquist, J.O., 2006. LS-DYNA Theoretical Manual. Livermore Software Technology Corporation, http://www.lstc.com/. HEEDS (Hierarchical Evolutionary Engineering Design System), 2008. Getting Started Manual. Red Cedar Technology, MI, USA, http://www.redcedartech.com/. Imaninejad, M., Subhash, G., Loukus, A., 2005. Loading path optimization of tube hydroforming process. Int. J. Mach. Tools Manuf. 45 (12–13), 1504–1514. Jansson, M., Nilsson, L., Simonsson, K., 2007. On process parameter estimation for the tube hydroforming process. J. Mater. Process. Technol. 190 (1–3), 1–11. Johnson, K., Nguyen, B., Davies, R., Grant, G., Khaleel, M., 2004. A numerical process control method for circular tube hydroforming prediction. Int. J. Plast. 20, 1111–1137. 2007. LS-DYNA Keyword Users Manual, Version 971. LSTC. Manabe, K., Suetake, M., Koyama, H., Yang, M., 2006. Hydroforming process optimization of aluminum alloy tube using intelligent control technique. Int. J. Mach. Tools Manuf. 46 (11), 1207–1211. Ni, C.-M., 1994. Stamping and hydroforming process simulation with a 3D finite element code. SAE Technical Paper 940753. Simo, J.C., Ortiz, M., 1985. A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations. Comput. Methods Appl. Mech. Eng. 49, 221–245. Stoughton, T.B., 2001. Stress-based forming limits in sheet metal forming. J. Eng. Mater. Technol. 123, 417–422. Stoughton, T., Zhu, X., 2004. Review of theoretical models of the strain-based FLD and their relevance to the stress-based FLD. Int. J. Plast. 20 (8–9), 1463–1486. Wu, L., Yu, Y., 1996. Computer simulations of forming automotive structural parts by hydroforming process. In: Proceedings of Numisheet’96. Yingyot, A.-U.-L., Gracious, N., Taylan Altan, 2004. Optimizing tube hydroforming using process simulation and experimental verification. J. Mater. Process. Technol. 146, 137–143.