Analytical and numerical modeling of high-strain rate metal-tube forming

  • S. GhadamiEmail author
  • B. Mollaei Dariani


In this research, a novel plasto-dynamic analytical and numerical modeling is developed for tube expansion during a high-strain rate forming process, such as electromagnetic and liquid shock tube forming. The analytical model is based on applying the impact mechanic and shell theories with introducing an effective load along with imposing high-order non-linearity of material behavior. The numerical model is also conducted by dynamic explicit finite element setup with considering an equivalent transient and traveling load. Both analytical and numerical models are compared ‌with each other and validated with experimental measurements of different processes available in literature. The analytical model has satisfactorily acceptable prediction, especially for small strains, locations far from dies or supports, and for processes with high load-to-material resistance ratio and with less complicated interaction and loading nature. The numerical modeling is also effective if precise data input is provided. Several parameter studies are carried out, and conditions for achieving a high-performance forming process are discussed. The magnitude of total impulse and inertia force have significant roles, but traveling speed and profile shape of load have no considerable effects on plastic deformation. For AISI 316L, strain hardening and thermal softening have major and minor influences on material strength, respectively.


High-strain rate forming High-speed forming Tube forming Analytical modeling Numerical simulation Dynamic explicit finite element method 

1 Introduction

High-energy rate metal-working processes are those in which a high-level energy is released in an extremely small time and are used mostly to form, weld, cut, harden, and compact the metals [1]. High-strain rate forming of tubes includes electromagnetic, electrohydraulic, liquid shock tube, gas detonation, and explosive forming. They have many advantages over traditional processes, like extended formability for some alloys, reduced tool cost, higher ability to form large and complex parts as well as difficult-to-form materials, better dimensional accuracy, and reduced force and energy because of inertia. So, they have attracted increased attention from manufacturers.

In these forming processes, the released energy acts as a forming tool and a high-level dynamic pressure is transferred by a midway medium to a tube. In some processes, this load has a traveling nature. This loading situation is followed by an elastic–viscoplastic deformation with large strain, high strain rate (usually about 103 to 105 s−1), high material velocity, and increased temperature in adiabatic condition [1, 2, 3]. This is regarded as severe loading. Laboratory studies of these processes necessitate special instruments. So, a combination of analytical and numerical models can be helpful and efficient.

Kleiner et al. [4] arranged a set of laboratory tube-free expansion tests by gas detonation forming with a controlled mix of oxygen and hydrogen. The mixed gas acted as energy source and as energy-transmitting medium too. They measured the radial displacement by high-speed photography technique and the pressure at two positions along the tube length by piezoelectric sensors. They reported shock wave reflection from different areas of the setup. Their numerical simulation had large errors, but it had good agreement with the experiment qualitatively. Also, they observed enhanced formability and better dimensional accuracy compared to the quasi-static hydroforming process. Psyk et al. [5] conducted electromagnetic-free tube expansion tests. They measured plastic deformation by a combination of high-speed photography and partial shadowing of a light beam. They reported time history of radial displacement and radial velocity and that of pressure. They investigated these tests as a basic process to reveal correlations between electromagnetic welding parameters to optimize the design, and they were successful. Jeanson et al. [6] tried to characterize high strain rate behavior of materials by electromagnetic tube expansion testing. Yu et al. [7], in a pure experimental work, concluded that energy utilization efficiency of the electrohydraulic tube bulging is higher than the electromagnetic process with the same deformation. Woo et al. [8] used the finite element method to simulate electrohydraulic forming process. Li et al. [9], in a pure numerical work, reported that simulation is essential to design a sophisticated electromagnetic forming process and to control its performance. Their numerical results were in good agreement with experiments of other’s research works. Zheng et al. [10] emphasized that because of coupling effect of gas–fluid–structure system and complex nature of the electrohydraulic tube forming, the experiment cannot be efficient alone and should be combined with numerical simulation. So, they used the finite element method to analyze dynamic plastic behavior of aluminum alloy under the action of shock wave loading in local bulging and concluded that FEM is in good agreement with the experiment. Qiu et al. [11] focused on common problem in high-strain rate tube-forming which is inhomogeneous deformation of tube or end effect. They used the finite element method to study the effect of coil shape in electromagnetic forming on output force to get a better force distribution. Ma et al. [12] reported that the inertia effect has significant role in electromagnetic plate forming. According to [13, 14, 15, 16], the material behavior at high strain rate is different from quasi static mode and constitutive equation for numerical simulation should be strain rate-sensitive. Some of them used temperature-insensitive material model and get satisfactory results. They used dynamic explicit solver.

To date, the research work on high-strain rate forming of tubes is rare, and most of them focused on electromagnetic forming and only have a combination of numerical and experimental methodologies. There are many open questions about process parameters. Here, the aim is to shed new light on these processes through the development of a simple and efficient analytical model and a reliable numerical simulation.

Free or die-less expansion of round tubes is a basic process in forming processes, which has been studied here. In this research, supposing a shell structure for tube and combining the membrane and impact mechanic theories, an analytical model based on cross-section method by two approaches, dynamic pressure loading (DPL) and impulsive loading (IL), was developed and a mathematical code was generated. Also, a reliable commercial finite element package was used as an extra tool to assess the analytical model and to process the study. The analytical and numerical models were validated by studying the instrumented works of Kleiner et al. [4] and Psyk et al. [5]. At last, some important process parameters were investigated.

2 Analytical modeling

The forming force in these processes is a transient pressure signal and, in some cases, a traveling and multistage load too. Considering a tube with schematic representation in Fig. 1a,b, the idealized and equivalent pressures for different points on the tube wall can be shown in Fig. 1c. So, the pressure can be described mathematically by Pforming = P0 × ϕ(x) × ψ(t), wherein ϕ(x) and ψ(t) are spatial (x) and time (t) functions and have an amount between zero and one, respectively. Also, P0 denotes the maximum pressure [17, 18]. These parameters are process-dependent. P0 usually has amount of several tens of MPa and, in some cases, can reach to GPa. ϕ(x) is a step function and depends on the position of each point and the shock wave velocity, which is mostly between about 1000 and 8000 m/s [1, 17, 19, 20, 21]. ψ(t) can usually be described as a smoothed exponential function [17, 18].
Fig. 1

Schematic of tube geometry and forming load. a Tube and its length along with r and x axes. b Thickness (H) and mean radius (R) of tube. c Idealized loading of two different points on tube wall along x axis with arrival times of t1 and t2

To develop an analytical model, some simplifying but reasonable assumptions must be made [15]. First of all, it is assumed that the pressure load acts simultaneously over the whole wall, which means that traveling speed is infinite and implies ϕ(x) = 1. This coincides well with processes in which there is no traveling load like electromagnetic forming or ones with very high shock wave velocity like explosive forming [1]. The time dependency of pressure is taken as an exponential function, i.e., Pforming = P0 × exp(−t/θ), wherein θ is time decay constant. The tube is taken as a shell structure, in which the membrane theory is valid [22]. Also, the longitudinal strains are neglected and it is assumed that the deformation is plane strain.

Although the stress–strain state is three-dimensional and flexural and shear forces along with the tension loads exist, for the sake of simplicity, moment loads along with shear loads and strains are neglected as a foundation of membrane condition. So, the remaining forces are as shown in Fig. 2a.
Fig. 2

a Basic concept of the cross-section method and applied forces for an infinitesimal element with an angle of . b Time history of tube expansion for a rigid-perfect plastic material showing the two phases encountered. c Schematic of three basic shapes of pressure signal and their characterization parameters

Therefore, based on the cross-section method, a simple ordinary differential equation (ODE) governs the dynamic plastic motion that is as follows [22]:
$$ -R\frac{d{Q}_x}{dx}-{N}_h+ RP=\rho HR\frac{\partial^2W}{\partial {t}^2}\&\frac{d{M}_x}{dx}={Q}_x $$
$$ {Q}_x\&{M}_x\cong 0\Rightarrow P(t)-\frac{N_h}{R}=\rho H\frac{\partial^2W}{\partial {t}^2}=\mu \frac{\partial^2W}{\partial {t}^2} $$

Whereas R is mean radius, W represents radial displacement, H thickness, ρ density, μ density in nature defined as μ = ρH, t time, and P(t) axisymmetric pressure. Mx, Qx, and Nh are generalized stresses. In other words, \( {M}_x={\int}_{-H/2}^{+H/2}{\sigma}_x dz \) is moment, \( {Q}_x={\int}_{-H/2}^{+H/2}{\sigma}_{xz} dz \) is lateral shear force, and \( {N}_h={\int}_{-H/2}^{+H/2}{\sigma}_h dz \) determines the membrane force or tension in the hoop direction. σ is stress, and the subscripts h, x, and z denote hoop, longitudinal, and thickness directions, respectively [22]. Constancy of σh along thickness results in Nh = H × σh. The term Nh/R is referred to material resistance force. From shell theory, σz ≅ 0; and, due to plastic flow rule of Levy–Mises for plane-strain condition, σh = 2σx. So, according to Tresca yield criterion, the equation σh = Y or Nh = N0 = Y × H governs the plastic yield where Y is the flow stress for a rigid-perfect plastic material. The required pressure to yield is PC = Y × H/R = N0/R.

Now, it is possible to solve the motion equation based on two different approaches, namely dynamic pressure loading (DPL) and impulsive loading. In DPL, the forming load is considered as a time-dependent function, which is applied directly to dynamic motion equations to solve and obtain time history of displacement and velocity. However, in the IL approach, the forming load is considered as an impulsive load. This is more accurate for loads with a large peak pressure to collapse pressure ratio (P0 ≫ Pc) and with very short duration (θ → 0). So, with a small time decay constant, it is assumed that the tube gets a uniform radial velocity instantaneously, which is calculated from linear momentum conservation. This impulsive velocity is only used to find the total forming time and final radial displacement.

Based on DPL approach and considering an idealized transient load, it suggests that tube expansion has two time phases as illustrated in Fig. 2b. At the first phase, the pressure is higher than the plastic flow limit and the tube accelerates from static. At the second phase, the pressure drops below the plastic flow threshold and only the inertia force is forming the tool and the tube continues to expand until the velocity becomes zero.

At the initial time of the first phase, both radial displacement and velocity are zero. The initial conditions of the second phase are those at the end of the first phase. The end time of the first phase, denoted by τ, can be derived as Pforming = Pc = P0 × exp(−τ/θ) ⇒ τ = θ × ln(Pc/P0).

So, for the second phase, the load is removed suddenly and Eq. (1) becomes a simple differential equation. Again, the variation of R and H was neglected. Also, the elastic deformation is ignored and the hoop tension is supposed to be constant, i.e., Nh = N0. At last, the solution for time-dependent relation of radial speed and displacement at two phases becomes
$$ \mu \frac{\partial {W}^I}{\partial t}={P}_0\theta \left(1-{\mathrm{e}}^{\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{$\theta $}\right.}\right)-\frac{N_0}{R}t $$
$$ {W}^I=\frac{P_0\theta }{\mu}\left(t+\theta {\mathrm{e}}^{\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{$\theta $}\right.}-\theta \right)-\frac{N_0}{2\mu R}{t}^2 $$
$$ \mu \frac{\partial {W}^{II}}{\partial t}={P}_0\theta \left(1-\frac{P_c}{P_0}\right)-\frac{N_0}{R}t $$
$$ {W}^{II}=\frac{P_0\theta }{\mu}\left(1-\frac{P_c}{P_0}\right)t-\frac{N_0}{2\mu R}{t}^2+\frac{P_0{\theta}^2}{\mu}\left[\frac{P_c}{P_0}\left(\mathit{\ln}\left(\frac{P_0}{P_c}\right)+1\right)-1\right] $$
Furthermore, the total forming time (tf) and final radial displacement of tube (Wfinal) are
$$ \frac{\partial {W}^{II}}{\partial t}=0\Rightarrow {t}_f=\frac{\theta \left({P}_0-{P}_c\right)}{P_c} $$
$$ {W}_{final}=\frac{P_0^2{\theta}^2R}{2\mu {N}_0}{\left(1-\frac{P_c}{P_0}\right)}^2+\frac{P_0{\theta}^2}{\mu}\left[\frac{P_c}{P_0}\left(\mathit{\ln}\left(\frac{P_0}{P_c}\right)+1\right)-1\right] $$
In these processes, the pulse shape of the pressure is not so smooth and also may be of forms rather than exponential. According to Fig. 2c, three basic forms with exponential (Ps1), linear (Ps2), and rectangular (Ps3) shapes with the same total impulse per unit area were studied. The impulse per unit area, denoted here as I, is time integration of a pressure signal from zero to a given time. To have the same value of impulse per unit area, for exponential pulse, the ψ(t) = exp(−t/θ) and τ = θ × ln(P0/Pc); for linear signal pulse, the ψl(t) = 1 − ((P0 + Pc)/(2θP0)) × t and τl = 2θ × ((P0 − Pc)/(P0 + Pc)); and for rectangular pulse, the ψr(t) = 1 and τr = θ × (1 − Pc/P0). Solving the motion equations for linear signal, the radial velocity and radial displacement equal to
$$ \mu \frac{\partial {W}^I}{\partial t}=\frac{1}{2{\tau}_l}\left(-{P}_0+\frac{N_0}{R}\right){t}^2+\left({P}_0-\frac{N_0}{R}\right)t $$
$$ {W}^I=\frac{1}{6\mu {\tau}_l}\left(-{P}_0+\frac{N_0}{R}\right){t}^3+\frac{1}{2\mu}\left({P}_0-\frac{N_0}{R}\right){t}^2 $$
$$ \mu \frac{\partial {W}^{II}}{\partial t}=-\frac{N_0t}{R}+\frac{N_0{\tau}_l}{2R}+\frac{P_0{\tau}_l}{2} $$
$$ {W}^{II}=\frac{N_0\left({\tau}_l^2-{t}^2\right)}{2\mu R}+\left(\frac{N_0{\tau}_l}{2\mu R}+\frac{P_0{\tau}_l}{2\mu}\right)\left(t-{\tau}_l\right)+\frac{1}{3\mu}\left({P}_0-\frac{N_0}{R}\right){\tau}_l^2 $$
Furthermore, for the rectangular pulse, the motion relations are
$$ \mu \frac{\partial {W}^I}{\partial t}=\left({P}_0-\frac{N_0}{R}\right)t $$
$$ {W}^I=\frac{1}{2\mu}\left({P}_0-\frac{N_0}{R}\right){t}^2 $$
$$ \mu \frac{\partial {W}^{II}}{\partial t}=-\frac{N_0}{R}t+{P}_0{\tau}_r $$
$$ {W}^{II}=-\frac{N_0}{2\mu R}{t}^2+\frac{P_0{\tau}_r}{\mu }t-\frac{P_0{\tau}_r^2}{2\mu } $$
However, based on the IL approach and assuming a very small time decay constant, the first phase is neglected. As mentioned before, the produced impulsive velocity is used as an initial condition for the second phase. The flow stress is supposed to have a mean value. According to the classic momentum rule, the velocity can be obtained as follows [22]:
$$ I={\int}_0^{\tau }{P}_0{e}^{\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{$\theta $}\right.}=\theta \left({P}_0-{P}_c\right) $$
$$ I=\mu \left(V-0\right)\Rightarrow V=\frac{I}{\mu } $$
Combining Eq. (6) and Eq. (2), the final radial displacement can be solved as:
$$ {t}_f=\frac{\mu V}{P_c}\Rightarrow {W}_{final}=\frac{\mu R{V}^2}{2{N}_0}+\frac{P_0{\theta}^2}{\mu}\left[\frac{P_c}{P_0}\left(\mathit{\ln}\left(\frac{P_0}{P_c}\right)+1\right)-1\right] $$
Traditional forming processes are isothermal. On the contrary, high-strain rate processes are adiabatic from a thermodynamics point of view and some of the generated heat cannot conduct and radiate, which raises the material’s temperature. There are two sources for temperature increase in the high-strain rate forming. One is the energy of plastic deformation, and another is the heat generated because of induced high-pressure shock wave in material or shock heating. The thermodynamic equation for temperature increment is the following [23, 24]:
$$ dT=d{T}_{plastic\ work}+d{T}_{shock\ heating}=\frac{\beta \times {\overline{\sigma}}_p\times d{\overline{\varepsilon}}_p}{\rho {c}_V}+ T\gamma d{\varepsilon}_V $$
where T represents temperature, γ Gruneisen and Hugoniot parameter, V volumetric strain increment, \( {\overline{\sigma}}_p \) equivalent plastic strain, \( d{\overline{\varepsilon}}_p \) equivalent plastic strain increment, cV specific heat, ρ density, and β the fraction of plastic work transferred to heat and usually about 0.9 [23, 24]. Again, for the sake of simplicity, the volumetric strain is taken as zero and the shock heating is neglected. Considering a plane-strain deformation and combining Eq. (8) and Eq. (2), the instantaneous temperatures of the two phases are as follows:
$$ {T}^I={T}_0+\frac{4\beta Y}{3 R\rho {c}_V}\left(\frac{P_0\theta }{\mu}\left(t+\theta {\mathrm{e}}^{\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{$\theta $}\right.}-\theta \right)-\frac{N_0}{2\mu R}{t}^2\right) $$
$$ {T}^{II}={T}_0+\frac{4\beta Y}{3 R\rho {c}_V}\left\{\frac{P_0\theta }{\mu}\left(1-\frac{P_c}{P_0}\right)\mathrm{t}-\frac{N_0}{2\mu R}{t}^2+\frac{P_0{\theta}^2}{\mu}\left[\frac{P_c}{P_0}\left(\mathit{\ln}\left(\frac{P_0}{P_c}\right)+1\right)-1\right]\right\} $$

The assumption that the stress is two phases is too rough and not fine enough. So, in order to achieve a precise model, the motion is divided to very small time steps. In each time step, the material strength and the load are updated and the motion parameters are solved.

Finally, based on the algorithm shown in Fig. 3, a code was developed by a mathematical package named MATLAB, in which with a series of small time steps, the motion is solved and strain hardening, strain rate hardening, and thermal softening are calculated by the Johnson–Cook constitutive equation according to Eq. (10) [25] for updating the flow stress to achieve a much more precise model.
Fig. 3

Flowchart of analytical modeling

$$ \overline{\sigma}=\left(A+B\ {\overline{\varepsilon}}^n\right)\left(1+C\ \mathit{\ln}\left(\dot{\overline{\varepsilon}}/\dot{{\overline{\varepsilon}}_0}\right)\right)\left[1-{\left(\left(T-{T}_{\mathrm{room}}\right)/\left({T}_{\mathrm{melt}}-{T}_{\mathrm{room}}\right)\right)}^m\right] $$

where \( \overline{\sigma} \) is effective flow stress, \( \overline{\varepsilon} \) effective plastic strain, \( \dot{\overline{\varepsilon}} \) effective plastic strain rate, \( \dot{{\overline{\varepsilon}}_0} \) reference plastic strain rate, T temperature, and Tmelt melting temperature. The parameters A, B, n, C, and m are experimentally determined material constants.

Any constitutive equation has its own application limits. The Johnson–Cook model is basically a phenomenon and was developed for special ranges of strain rate and temperature. The suggested range for the strain rate was 103–105 s−1, and the temperature should be small enough to assure that recrystallization does not occur [25]. Recrystallization is a metallurgical phenomenon and a kind of softening mechanism. It is usually important at high temperatures of about 0.5–0.7 Tm (where Tm is melting temperature in Kelvin), i.e., at hot-forming processes or cold-forming of some pure metals.

In this study, the maximum strain rate was about 104 s−1. Peak temperature for the studied materials was below of the threshold. So, in this research work, the strain rates and temperature are at the specified range and the Johnson–Cook model can be used safely.

xThis constitutive equation does not take into account the microstructural effects, but it is sufficient to predict the tube motion herein.

In this work, the main strategy in modeling is to make some simplifying and reasonable assumptions to avoid complex equations to get a simple and easily applicable formula. Thus, it is assumed that the considered surface load is an idealized one and is equivalent for any type of load which may exist. For example, in electromagnetic forming, the load is a body force. It can be simply idealized as a surface load. Due to thin thickness of the tube, this assumption seems to be reasonable.

3 Numerical simulation (finite element analysis)

In this research, the powerful commercial finite element (FE) package, ABAQUS, was used for numerical simulation. Due to the short duration of the dynamic plastic deformation and high material’s non-linearity, the Dynamic/Explicit solver was employed. In spite of full axisymmetric geometry and loading condition, a 3D model was developed. The dies, supports, or any tools, if existed, were modeled as rigid bodies because they do not experience any plastic deformation. The C3D8R element that is an 8-node linear brick with reduced integration and hourglass control was selected. To optimize selection of mesh density and assure that it has no bad effect on the results, a mesh sensitivity and convergence study was conducted. The optimum value of the approximate global size was obtained as 2 mm, and 40 circumferential elements were selected. The forming load was applied as an idealized transient pressure of the required profile on the tube’s inside wall. In some simulations, the load was applied simultaneously to all points, and, in some cases, it was modeled as a traveling pressure along the tube wall. The time history of pressure can be obtained from some theoretical models or be measured by special transducers [4]. Supposing an isotropic material, the von Mises yield criterion along with the Johnson–Cook constitutive equation was used to model the material flow stress considering work hardening and softening features. Also, elastic behavior was included. A sample FEM model is illustrated in Fig. 4.
Fig. 4

Finite element model of a sample tube-free forming system with two end supports

4 Results and discussion

First of all, in order to validate the numerical simulation and analytical model and to examine their performance, the well-instrumented laboratory tube-free expansion tests which were done by Psyk et al. [5] and Kleiner et al. [4] were studied. The material data is listed in Table 1. It is worth noting that for AISI 316L, the parameters A, B, and n were determined by the authors of this work via quasi-static uniaxial tension tests and subsequent least-square fitting of true stress–strain curve.
Table 1

Material data



AISI 316L [27]

EN AW-1050 or Al99.5 [26]

E, Elastic modulus (GPa)



ν, Poisson’s ratio



ρ, Density (kg/m3)



A, Material constant (MPa)



B, Material constant (MPa



n, Strain-hardening exponent



C, Material constant



m, Temperature sensitivity



\( \dot{{\overline{\varepsilon}}_0} \), Reference strain rate (s−1)



Tmelt, Melting temperature (°K)



Troom, Room temperature (°K)



cV, Specific heat (J × kg/°K)



From a manufacturing point of view, precise prediction of the displacement or plastic strain is more important than velocity or strain rate. Total forming time is the least important parameter.

For numerical modeling of the experimental tests of references [4, 5], three-dimensional elastic-plastic finite element model like that in Fig. 4 was arranged. For analytical modeling, based on Eqs. (2)–(10) and the flowchart in Fig. 3, a code was developed. The material and geometrical data of tubes and load parameters are listed in Tables 1 and 2. Also, important input parameters of FEM setup and analytic code are included.
Table 2

Modeling data and errors of modeling


Electromagnetic forming [5]

Gas detonation forming [4]


EN AW-1050

EN AW-1050

Thickness (mm)



Tube outer diameter (mm)



Tube length (mm)



Number of circumferential elements



Approximate global size of elements (mm)



Measurement position

Middle of tube

29 mm from supports

Time step for analytical model (μs)



P0, Peak pressure (MPa)



θ, Time decay constant (μs)



Traveling speed of load (m/s)



Pc, Static collapse pressure (MPa)



P0/Pc ratio



Error for displacement @100% of reported time, Analytical modeling

− 15%

− 29%

Error for displacement @100% of reported time, Numerical simulation

− 5%

− 24%

Error for displacement @40% of reported time, Analytical modeling

+ 2%

− 27%

Error for displacement @40% of reported time, Numerical simulation

− 2%

− 20%

Error for peak velocity, Analytical modeling

− 2%

Error for peak velocity, Numerical simulation

− 17%

4.1 Validation of analytical modeling and numerical simulation by experimental data of electromagnetic and detonation-forming processes

Figure 5a,b shows the comparison between the experimental measurement [5] and the results of theoretical models for radial displacement and velocity in electromagnetic tube-forming, respectively. The material and geometrical properties of the tube are listed in Table 2.
Fig. 5

Comparison of analytical modeling and numerical simulation (FEM) results with experimental measurement in electromagnetic tube forming (elastic vibration is not included) [5]. a Radial displacement history and b radial velocity history

The whole forming process lasted about 100 μs. The maximum radial velocity at the middle of the tube was about 185 m/s, which is equivalent to a plastic strain rate of about 9500 s−1.

The errors of the theoretical models, as compared to measurement data, are listed in Table 2. In spite of intricate forming situation, the numerical simulation predicted the dynamic plastic deformation well and has given a good fit to measurements of both parameters, especially for displacement. The analytical model with DPL approach had good agreement with experimental data and underestimated the motion parameters. It could predict well the peak radial velocity. The analytical results with impulsive loading approach were also included, which are higher than the measurement data.

According to Fig. 5a, the tube reached peak velocity at about 23% of the total forming time. The forming pressure dropped under initial yield strength limit (PC) at about 25% of the total time, and consequently about 75% of the process was due to inertia force.

In addition, according to the analytical model, ignoring strain hardening led to about 128 and 19% overestimation of the final displacement and peak velocity, respectively. For neglecting the strain rate effect, these were overestimated about 9 and 3%, respectively. Ignoring temperature impact resulted in about 8 and 2% underestimation of these parameters, respectively.

Figure 6 compares the experimental data of gas detonation tube-forming [4] to the results of numerical and analytical models for radial displacement. The errors of theoretical models, as compared to measurement data, are listed in Table 2.
Fig. 6

Comparison of analytical modeling and numerical simulation (FEM) with experimental measurement in gas detonation forming [4] for radial displacement history at 29 mm distance from end ring

Also, according to the FEM analysis, at the middle area, the mean ratio of longitudinal plastic strain to total effective plastic strain is about 1:8, which indicates that plane-strain state is valid approximately.

The calculation errors are higher as compared to the electromagnetic forming. Figure 7a–d shows the FEM output for the progress of this detonation forming at different times. Forming force is a high pressure and sharp-fronted shock wave along with the Taylor wave which travels with a traveling speed of about 3000 m/s [4]. For a given time, the numerical model underestimated displacement with an amount of about 15 to 24%. As it is clear, ignoring the moving nature and applying the load simultaneously over the whole wall resulted in more error that was up to 30%. The analytical model predicted the displacement with about 29% error. The experimental data was only up to about 60 μs which tube did not rest [4].
Fig. 7

FEM output for the progress of gas detonation tube-forming process at different times and developed von Mises effective plastic strain (with ×4 magnification of deformation): a 0, b 54, c 108, and d 162 μs

From the above-mentioned comparisons, it can be concluded that both numerical and analytical predictions had acceptable agreement with the experimental data. The two theoretical tools could predict the motion parameters much better for electromagnetic forming as compared to gas detonation forming. Possible reasons for these errors can be explained in the following:
  1. a)

    To achieve an accurate theoretical prediction, it is necessary to have a correct calculation of the material resistance force. Exact material data, suitable constitutive equation, proper yield criterion, and acceptable estimation of stress–strain state are main pre-requirements. Moreover, the effects of thickness (H) and radius (R) variations which decrease the resistance should be calculated correctly.

    Unfortunately, the proposed analytic model has some simplifying assumptions which are potential sources of error. For example, the constancy of H and R leads to overestimation of resistance force. Also, the Tresca yield criterion does not include the effects of all normal and shear stresses. In addition, assuming a plane-strain state is not suitable especially for large strains and areas near ends, dies, or supports. The plane-strain state and ignoring of longitudinal strains can result in high-resistance force. Ignoring of flexural waves is another important feature.

  2. b)

    Owing to the fact that the stress–strain state at the middle area of the tube is plane strain and coincides well with the assumptions of analytic model, it can be said surly that if the measurement position was at the middle of the tube, the prediction would be much better.

    As can be seen in Fig. 7d, the displacement was unequal for different longitudinal positions. It was almost uniform at middle areas, but there was a rise near the two end rings. The analytic model is not sensitive to this phenomenon and predicts a unique value for all locations. Kleiner et al. [4] pointed out the above-mentioned behavior but measured only the deformation of two points located at 29 mm from the ends. They did not measure the deformation of the middle area. Therefore, it was not possible to compare the data.

  3. c)

    The assumptions of both analytic and FEM models get progressively worse at high plastic strains and lead to high errors.

  4. d)

    In the electromagnetic formation [5], the pressure was applied in a straightforward manner approximately and the load was not a traveling one. Also, there was almost no special interaction and contact phenomenon between the tube wall and the transmitting medium, which was air.

    By contrast, in detonation forming [4], the load was a traveling pressure and the transmitting medium was a combined mix of oxygen and hydrogen gases which induced complicated interaction and contact phenomenon to the tube. Also, the reflection of shock waves from different areas of forming setup, especially from two ends, occurred [4]. The numerical simulation and analytical model could not reproduce these complex phenomena accurately. So, the complexity of interactions in the energy source–medium–tube system is very important.

  5. e)

    As shown in Table 2, the ratio of forming load to material resistance (which can be approximated as P0/Pc ratio) is higher in the electromagnetic forming as compared to the detonation forming, and the time decay constant is vice versa. This means that in the detonation forming [4], material resistance, which is not accurately predictable, plays more important role.

  6. f)

    Due to lack of information, the magnifying effects of secondary and following loads were not considered in the analytical and numerical modeling. Because of shock wave reflection, the load has a multi-stage character in detonation forming. In reference [4], the reflection was pointed out, but one step was only reported and data was not complete. So, there was a lack of input data for the load. This deficiency led to high errors in both analytical and numerical modeling.

  7. g)

    The amplifying effects of advancing flexural waves which develop deformation at each point before the arrival of detonation are important. The analytic model did not include this feature [17, 18].

  8. h)

    Some uncertainties from material property data may be another source of error, especially for dynamic loading which, rather than static tests, some especial high speed tests are needed to characterize the material behavior.

  9. i)

    Like any other laboratory studies, there were some errors in the experimental data of references [4, 5], which caused errors in this study.

  10. j)

    The analytic and numerical models use the Tresca and von Mises yield criterions, respectively. This can be a source for errors between these two models which leads a maximum error of about 15% for the yield stress. The Tresca yield criterion is also called the “maximum shear stress” criterion and can be expressed as σ1 − σ3 = Y (with the convention σ1 ≥ σ2 ≥ σ3, wherein σ1, σ2, and σ3 are principle stresses and Y is flow stress). It does not include the terms of all normal and shear stresses. In contrary, the von Mises criterion has more complicated form and includes all normal and shear stresses. The von Mises model is used widely with finite element modeling in literature to calculate flow stress and effective stress. Herein, due to applicability of Tresca criterion, it was used in analytical modeling.


It is worth mentioning that regarding the advantages of high-speed forming with respect to quasi-static tube forming, it was realized that in high-speed processes, strain rate and material velocity are higher and inertia has much more substantial role. So, strain rate hardening can lead to produce parts with high strength. Also, in rate-sensitive materials, the formability can be improved. Inertia force acts as a forming tool, which is useful in forming large parts and difficult-to-form materials. This means energy storage. Also, inertia can improve dimensional accuracy of fabricated tube. The high velocity of the material can be used to weld two tubes to produce two-walled parts.

In the following studies, a tube made of AISI 316L with mean diameter of 19.3 mm, thickness of 1.4 mm, and length of 100 mm was used.

4.2 Comparison of analytical modeling and numerical simulation (FEM)

Figure 8a–c compares the computed results of analytical modeling and numerical simulation for different values of impulse per unit area. For a same impulse, analytical model underestimated the displacement, and the higher the impulse the more gap arises between the two models. The displacement error for the impulse of 2800 N s/m2 is about 15%. This error can be described according to the fact that as the deformation increases, the assumptions of the analytic model are ignored more and more. For maximum radial velocity and peak strain rate, the variation is much lower and is less than about 2%. It is due to the fact that the tube attains its peak velocity and strain rate at initial times of forming, for example at 7 μs, as compared to total forming time of 59 μs for 2400 N s/m2, in which deformation is not large. It is worth noting that according to the analytical model, about 80 to 85% of these processes were due to inertia force.
Fig. 8

Comparison of analytical and numerical results for different impulses per unit area; a final radial displacement, b peak radial velocity, c total forming time

As can be seen in Fig. 9a,b, if a constant impulse is applied as faster as possible (with lower value of θ or higher value of P0), the analytic model and numerical simulation get closer to each other. In fact, for high levels of pressure, the material resistance, which analytic model cannot compute well, has little effect on motion. Therefore, the analytical model can predict well processes with higher peak pressure or short time decay constants.
Fig. 9

Comparison of analytical and numerical results for different values of time decay constant with I = 2400 N s/m2; a final radial displacement, b peak radial velocity

4.3 Comparison of impulsive loading and dynamic pressure loading approach

Figure 10 compares the results of DPL approach to that of IL for different values of impulse per unit area. The energy-based mean yield stress with the Johnson–Cook constitutive equation was used in the IL. The IL predicted higher values for the final displacement. The difference was more for the higher values of impulse. This can be attributed to the fact that impulsive loading calculates higher peak velocity. As an example, for impulse per unit area of 1600 and 2800 N s/m2, the peak velocities were about 142 and 250 m/s with IL as compared to 114 and 214 m/s with DPL. Also, the relation between total impulse and final displacement is not linear.
Fig. 10

Analytical comparison of impulsive loading (IL) and dynamic pressure loading (DPL) approach

Figure 11 shows the effect of material parameters versus different amounts of impulse per unit area obtained by the analytical method. Ignoring the strain, strain rate, or temperature effects leads to about 77, 40, or 5% average difference in the results, respectively. So, strain hardening has major impact on material strength and thermal softening has minor effect.
Fig. 11

Analytical comparison of final radial displacement for DPL and IL approach (different material model included)

4.4 Effect of pressure pulse characteristics

Figure 12 shows the analytical results for a unique impulse per unit area (I = 2400 N s/m2) with different combination of time decay constant and maximum pressure from θ= 1 s (P0 = 2.42 GPa) to θ = 2.5 s (P0 = 0.98 GPa). The motion parameters decrease with time decay constant.
Fig. 12

Analytical modeling of tube wall deformation with different values of time decay constant (I = 2400 N s/m2)

So, in order to increase the efficiency, in addition to the improvement of energy utilization factor, forming force should be applied as fast as possible. Another practical interpretation is that by taking some measures the efficiency can be improved which some of them are the using of a suitable transferring medium, intensifying the load by shock wave-focusing devices (like shock reflectors), or increasing confinement at some locations [1, 19, 20, 21]. Also, in most of high-speed tube-forming processes, the so-called end effect problem is inherent which is due to the lack of suitable pressure loading at the two ends, for example, because of gas escape. This problem can be minimized by manipulating the impulse at those areas.

It is worth mentioning that there are different materials for transferring medium, such as water, sand, polyethylene, and air. The most important characteristics of medium are density, sound speed, compressibility, permeability, and interaction property, which can be selected in an optimum manner to get higher effective plastic strain without necking or cracking [1, 3]. Cavitation, tearing, and melting of the medium are some important observed interaction problems [1].

Figure 13a,b shows the analytical model’s prediction for the effects of pressure pulse shape. Overall, the efficiency of forming is highest for the rectangular pulse and lowest for the exponential one. For small θ, which equals to high P0, the pulse shape has no significant effect on the parameters.
Fig. 13

Analytical analysis of pulse shape effects. a The impact of pulse shape on tube wall motion. b Final radial displacement for different values of time decay constant at a constant impulse (I = 2400 N s/m2)

This observation is encouraging from a practical viewpoint. This finding means that the total impulse of a load is more important than its shape, especially in processes with very low θ. Also, in some processes, there are wave reflections from different positions of setup which changes the shape and applies secondary loads. It is difficult to fit a unique mathematical function to the signal, so that it can be approximately idealized with an effective load with the same total impulse.

Another important feature is the traveling speed. Figure 14a,b depicts the numerical simulation results for different speeds of traveling pressure comparatively. Totally, there is a slight reverse relation between traveling speed and motion parameters. By changing from 1000 to 10,000 m/s, only 3% reduction was observed. It can be said that as the speed increases, tube material does not have enough time to flow longitudinally and material flow in radial direction is retarded. In other words, the plane-strain state becomes dominant.
Fig. 14

FEM analysis of deformation parameters with respect to traveling speed of pressure load; a final radial displacement, b peak radial velocity (I = 2400 N s/m2 and θ = 2 μs)

For the non-moving case, the maximum displacement and peak radial velocity were 4.82 mm and 180 m/s, respectively, which are lower than those of the moving condition. Thus, traveling has an amplifying effect. This increase to some extent can be attributed to the development of flexural waves.

It is worth noting that these findings must be interpreted with caution, because for the elastic response, a resonance effect was reported at a very narrow range of speed which was attributed to the coupling between flexural wave and detonation loading [17, 18].

4.5 Influence of length-to-diameter ratio of tube

Figure 15a,b presents the effect of length-to-diameter ratio of tube on the main plastic deformation parameters which were calculated by FEM analysis. As anticipated, a negative correlation was found between the radial displacement and the ratio. One main reason is that at the two ends of the tube the deformation state is alike uniaxial tension and it gradually shifts toward plane-strain state upon moving to the middle area. In other words, in long tubes, there is a higher longitudinal resistance and the material resistance is higher.
Fig. 15

FEM analysis of the effect of length-to-diameter ratio on motion parameters (I = 2400 N s/m2 and θ = 2 μs); a final radial displacement, b peak radial velocity

The peak velocity (and the maximum strain rate) remained constant approximately. This is due to the fact that the tube gets its maximum velocity at early times of motion when pressure is too higher than the material resistance force. Thus, at early times, material resistance has no considerable role.

5 Conclusions

The high-strain rate die-less tube-forming was modeled and investigated by a combination of analytical modeling and finite element method. Their adequacy was validated too. The main conclusions are as follows:
  1. 1.

    The proposed analytical model is effective and applicable. It is more accurate in prediction of very fast processes with high load-to-material resistance ratio and with less interaction phenomena, such as electromagnetic forming. It is useful for locations far from ends, dies, or supports where plane strain prevails. It is more general and faster than numerical simulation, and the parameter study is quite simpler.

  2. 2.

    Dynamic pressure loading approach underestimates motion parameters, but impulsive loading approach overestimates them.

  3. 3.

    The numerical simulation (FEM) has reasonable estimation except where the information of forming load is not available due to the complicated nature of the process interactions or lack of measurement data.

  4. 4.

    Magnitude of forming load’s total impulse, time decay constant, and inertia force have considerable roles. Pressure pulse shape has no significant impact on motion parameters, especially for low amounts of time decay constant.

  5. 5.

    Traveling speed of load and motion parameters show a limited reverse correlation. There is a negative relation between motion parameters and length-to-diameter ratio of tube.

  6. 6.

    A selection of an energy source and medium which provides a load profile with higher peak pressure or lower time decay constant leads to higher performance.

  7. 7.

    For AISI 316L, strain hardening and thermal softening have major and minor effects on material strength, respectively.

  8. 8.

    Further experimental investigations are needed to prove correctness of pure theoretical findings.




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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentAmirkabir University of TechnologyTehranIran

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